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Series Author and Senior Consultant
Marian Small
Lead Author
Chris Kirkpatrick
Authors
Mary Bourassa • Crystal Chilvers • Santo D’Agostino
Ian Macpherson • John Rodger • Susanne Trew
Principles of Mathematics 10
Series Author and Senior
Consultant
Marian Small
Lead Author
Chris Kirkpatrick
Vice President, Publishing
Janice Schoening
General Manager, Mathematics,
Science, and Technology
Lenore Brooks
Publisher, Mathematics
Colin Garnham
Associate Publisher, Mathematics
Sandra McTavish
Managing Editor, Mathematics
Erynn Marcus
Product Manager
Linda Krepinsky
Program Manager
Lynda Cowan
Developmental Editors
Amanda Allan; Nancy Andraos;
Shirley Barrett; Tom Gamblin;
Wendi Morrison, First Folio
Resource Group, Inc.; Bob
Templeton, First Folio Resource
Group, Inc.
COPYRIGHT © 2010 by Nelson
Education Limited.
ISBN-13: 978-0-17-633202-0
ISBN-10: 0-17-633202-2
Printed and bound in Canada
2 3 4 5 12 11 10 09
For more information contact
Nelson Education Ltd.,
1120 Birchmount Road, Toronto,
ON, M1K 5G4.
Or you can visit our Internet site at
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Authors
Mary Bourassa, Crystal Chilvers,
Santo D’Agostino, Ian Macpherson,
John Rodger, Susanne Trew
Technology Consultant
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Contributing Authors
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Reviewers and Advisory Panel
Paul Alves
Department Head of Mathematics
Stephen Lewis Secondary School
Peel District School Board
Mississauga, ON
Patricia Kehoe
Itinerant Teacher, Student Success
Department
Ottawa Catholic School Board
Ottawa, ON
Anthony Arrizza
Department Head of Mathematics
Woodbridge College
York Region District School Board
Woodbridge, ON
Michelle Lang
Consultant, Learning Services 7–12
Waterloo Region District School Board
Kitchener, ON
Terri Blackwell
Secondary Mathematics Teacher
Burlington Central High School
Halton District School Board
Burlington, ON
Mark Cassar
Principal
Holy Cross Catholic Elementary School
Dufferin-Peel Catholic District School Board
Mississauga, ON
Angela Conetta
Mathematics Teacher
Chaminade College School
Toronto Catholic District School Board
Toronto, ON
Tamara Coyle
Teacher
Mother Teresa Catholic High School
Ottawa Catholic School Board
Nepean, ON
Justin de Weerdt
Mathematics Department Head
Huntsville High School
Trillium Lakelands District School Board
Huntsville, ON
Sandra Emms Jones
Math Teacher
Forest Heights C.I.
Waterloo Region District School Board
Kitchener, ON
Beverly Farahani
Head of Mathematics
Kingston Collegiate and Vocational Institute
Limestone District School Board
Kingston, ON
Richard Gallant
Secondary Curriculum Consultant
Simcoe Muskoka Catholic District School
Board
Barrie, ON
Jacqueline Hill
K–12 Mathematics Facilitator
Durham District School Board
Whitby, ON
NEL
Angelo Lillo
Head of Mathematics
Sir Winston Churchill Secondary School
District School Board of Niagara
St. Catharines, ON
Susan MacRury
Senior Mathematics Teacher
Lasalle Secondary School
Rainbow District School Board
Sudbury, ON
Frank Maggio
Department Head of Mathematics
Holy Trinity Catholic Secondary School
Halton Catholic District School Board
Oakville, ON
Peter Matijosaitis
Retired
Toronto Catholic District School Board
Kathy Pilon
Program Leader
St. John Catholic High School
Catholic District School Board of Eastern
Ontario
Perth, ON
Jennifer Portelli
Teacher
Holy Cross Catholic Elementary School
Dufferin-Peel Catholic District School Board
Mississauga, ON
Tamara Porter
Department Head of Mathematics
Prince Edward Collegiate Institute
Hastings and Prince Edward District School
Board
Picton, ON
Margaret Russo
Mathematics Teacher
Madonna Catholic Secondary School
Toronto Catholic District School Board
Toronto, ON
Scott Taylor
Department Head of Mathematics, Computer
Science and Business
Bell High School
Ottawa-Carleton District School Board
Nepean, ON
Bob McRoberts
Head of Mathematics
Dr. G W Williams Secondary School
York Region District School Board
Aurora, ON
Joyce Tonner
Educator
Thames Valley District School Board
London, ON
Cheryl McQueen
Mathematics Learning Coordinator
Thames Valley District School Board
London, ON
Salvatore Trabona
Mathematics Department Head
Madonna Catholic Secondary School
Toronto Catholic District School Board
Toronto, ON
Kay Minter
Teacher
Cedarbrae C.I.
Toronto District School Board
Toronto, ON
Reshida Nezirevic
Head of Mathematics
Blessed Mother Teresa C.S.S
Toronto Catholic District School Board
Scarborough, ON
Elizabeth Pattison
Mathematics Department Head
Grimsby Secondary School
District School Board of Niagara
Grimsby, ON
James Williamson
Teacher
St. Joseph-Scollard Hall C.S.S
Nipissing-Parry Sound Catholic District School
Board
North Bay, ON
Charles Wyszkowski
Instructor
School of Education, Trent University
Peterborough, ON
Krista Zupan
Math Consultant (Numeracy K–12)
Durham Catholic District School Board
Oshawa, ON
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Table of Contents
Chapter 1: Systems of Linear Equations 2
Chapter 3: Graphs of Quadratic
Relations
128
Getting Started
4
1.1 Representing Linear Relations
8
Getting Started
130
1.2 Solving Linear Equations
15
3.1 Exploring Quadratic Relations
134
1.3 Graphically Solving Linear Systems
21
3.2 Properties of Graphs of Quadratic Relations 138
Curious Math
29
Mid-Chapter Review
30
1.4 Solving Linear Systems: Substitution
33
1.5 Equivalent Linear Systems
Curious Math
149
3.3 Factored Form of a Quadratic Relation
150
Mid-Chapter Review
159
41
3.4 Expanding Quadratic Expressions
161
1.6 Solving Linear Systems: Elimination
49
3.5 Quadratic Models Using Factored Form
169
1.7 Exploring Linear Systems
57
179
Chapter Review
60
3.6 Exploring Quadratic and
Exponential Graphs
Chapter Review
183
Chapter Self-Test
64
Chapter Self-Test
187
Chapter Task
65
Chapter Task
188
Chapters 1–3 Cumulative Review
189
Chapter 4: Factoring Algebraic
Expressions
192
Chapter 2: Analytic Geometry:
Line Segments and Circles
66
Getting Started
68
2.1 Midpoint of a Line Segment
72
2.2 Length of a Line Segment
81
Getting Started
194
2.3 Equation of a Circle
88
4.1 Common Factors in Polynomials
198
Mid-Chapter Review
94
4.2 Exploring the Factorization of Trinomials
205
2.4 Classifying Figures on a Coordinate Grid
96
4.3 Factoring Quadratics: x 2 + bx + c
207
2.5 Verifying Properties of Geometric Figures
104
Mid-Chapter Review
214
2.6 Exploring Properties of Geometric Figures
111
4.4 Factoring Quadratics: ax 2 + bx + c
217
114
4.5 Factoring Quadratics: Special Cases
225
Curious Math
2.7 Using Coordinates to Solve Problems
115
Chapter Review
122
4.6 Reasoning about Factoring Polynomials
233
Chapter Self-Test
126
Chapter Review
238
Chapter Task
127
Chapter Self-Test
242
Chapter Task
243
iv
Table of Contents
Curious Math
232
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Chapter 5: Applying Quadratic
Models
Page v
7.2 Solving Similar Triangle Problems
382
244
Mid-Chapter Review
389
Getting Started
246
7.3 Exploring Similar Right Triangles
391
5.1 Stretching/Reflecting Quadratic Relations
250
7.4 The Primary Trigonometric Ratios
394
7.5 Solving Right Triangles
400
Curious Math
407
5.2 Exploring Translations of Quadratic Relations 259
5.3 Graphing Quadratics in Vertex Form
263
Mid-Chapter Review
273
5.4 Quadratic Models Using Vertex Form
275
5.5 Solving Problems Using Quadratic Relations 285
Curious Math
296
5.6 Connecting Standard and Vertex Forms
297
Chapter Review
303
Chapter Self-Test
306
Chapter Task
307
7.6 Solving Right Triangle Problems
408
Chapter Review
415
Chapter Self-Test
418
Chapter Task
419
Chapter 8: Acute Triangle
Trigonometry
420
Getting Started
422
8.1 Exploring the Sine Law
426
Chapter 6: Quadratic Equations
308
8.2 Applying the Sine Law
428
Getting Started
310
Mid-Chapter Review
435
6.1 Solving Quadratic Equations
314
8.3 Exploring the Cosine Law
437
6.2 Exploring the Creation of Perfect Squares
322
Curious Math
6.3 Completing the Square
324
325
Mid-Chapter Review
333
6.4 The Quadratic Formula
336
6.5 Interpreting Quadratic Equation Roots
345
6.6 Solving Problems Using Quadratic Models
352
Chapter Review
439
Curious Math
8.4 Applying the Cosine Law
440
8.5 Solving Acute Triangle Problems
446
Chapter Review
452
Chapter Self-Test
454
Chapter Task
455
360
Chapters 7–8 Cumulative Review
456
Chapter Self-Test
363
Chapter Task
364
Appendix A:
REVIEW OF ESSENTIAL SKILLS AND KNOWLEDGE
459
Chapters 4–6 Cumulative Review
365
Appendix B:
REVIEW OF TECHNICAL SKILLS
486
Glossary
528
Chapter 7: Similar Triangles
and Trigonometry
368
Answers
536
Getting Started
370
Index
600
7.1 Congruence and Similarity in Triangles
374
Credits
604
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Chapter
1
Systems
of Linear
Equations
GOALS
You will be able to
• Solve a system of linear equations using
a variety of strategies
• Solve problems that are modelled by
linear equations or systems of linear
equations
• Describe the relationship between the
Cost of bulb and
electricity ($)
number of solutions to a system of linear
equations and the coefficients of the
equations
40
Comparing Light Bulb Costs
y
30
? Why does it make sense to buy
20
10
0
x
10000
5000
Hours of use
energy-efficient compact
fluorescent light bulbs, even
though they often cost more
than incandescent light bulbs?
incandescent light bulb
compact fluorescent
light bulb
NEL
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Getting Started
WORDS YOU NEED to Know
1. Complete each sentence using one or more of the given words.
Each word can be used only once.
i) x-intercept
v) coefficient
ii) y-intercept
vi) point of intersection
iii) equation
vii) solution
iv) variable
a) The place where a graph crosses the x-axis is called the _____.
b) In the _____ y 5x 2, 5 is a _____ of the _____ x.
c) Let x 0 to determine the _____ of y 4x 7.
d) You can determine the _____ to 20 3x 10 by graphing
y 3x – 10.
e) The ordered pair at which two lines cross is called the _____.
SKILLS AND CONCEPTS You Need
Graphing a Linear Relation
Study
Aid
• For more help and
practice, see
Appendix A-6 and A-7.
You can use different tools and strategies to graph a linear relation:
• a table of values
• the x- and y-intercepts
• the slope and y-intercept
• a graphing calculator
EXAMPLE
Graph 3x 2y 9.
Solution
Using the x- and y-intercepts
Let y 0 to determine the x-intercept.
3x 2(0) 9
3x 9
x3
The graph passes through (3, 0).
Let x 0 to determine the y-intercept.
3(0) 2y 9
2y 9
y 4.5
y
(0, 4.5)
5
4
3
2
1
-1
(3, 0) x
0
-1
-2
1
2
3
4
3x 2y 9
The graph passes through (0, 4.5).
Plot the intercepts, and join them with
a straight line.
4
Getting Started
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Getting Started
Using the Slope and y-intercept
3x + 2y = 9
2y = - 3x + 9
2y
- 3x
9
=
+
2
2
2
y = - 1.5x + 4.5
The slope is 1.5. The y-intercept is 4.5, so the line passes through (0, 4.5).
Plot (0, 4.5). Use the rise and run to locate a second point on the line,
by going right 1 unit and down 1.5 units to (1, 3).
2. Graph each relation using the slope and y-intercept.
a) y = 4x - 7
b) x + 2y = 3
y
(0, 4.5)
5
4
(1, 3)
3
2
1
x
-1
0
-1
-2
1
2
3
4
3x 2y 9
3. Graph each relation using the x- and y-intercepts.
a) 4x - 5y = 10
b) y = 2 - 3x
4. Graph each relation using the strategy of your choice.
a) x - 3y = 6
b) y = 5 - 2x
Expanding and Simplifying an Algebraic Expression
You can use an algebra tile model to visualize and simplify an expression.
If the expression has brackets, you can use the distributive property
to expand it. You can add or subtract like terms.
EXAMPLE
Study
Aid
• For more help and
practice, see
Appendix A-8.
Expand and simplify 2(3x 1) 3(x 2).
Solution
Using Symbols
Using an Algebra Tile Model
2(3x 1) 3(x 2)
1
1
x
x
x
x
x
x
1 1
1 1
1 1
x
x
x
1 1
1 1 1 1 1 1
x
x
x
x
x
x
2(3x 1) 3(x 2)
6x 2 3x 6
6x 3x 2 6
9x 4
x
x
x
9x 4
5. Expand and simplify as necessary.
a) 5x 10 3x 12
b) 4(3x 5)
c) 2(5x 2)
NEL
d) (3x 6) (2x 7)
e) 6(2x 4) 3(2x 1)
f ) (8x 14) (7x 6)
Chapter 1
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PRACTICE
Aid
• For help, see the Review
6. Rearrange each equation to complete the table.
of Essential Skills and
Knowledge Appendix.
Ax By C 0 Form
Question
Appendix
6, 7
A-7
10
A-10
11, 12
A-9
a)
y mx b Form
3x 4y 6 0
y 2x 5
b)
c)
4x 7y 3 0
d)
y = -
2
5
x 3
6
7. State the slope and y-intercept of each relation. Then sketch the graph.
a) y = 3x - 5
2
b) y = - x + 1
3
c) y = 0.5x
d) y = 2.6x - 1.2
8. Which relations in question 7 are direct variations? Which are partial
Distance from home (km)
variations? Explain how you know.
6
Kyle’s Journey Home
from School
y
9. The graph at the left shows Kyle’s distance from home as he cycles
home from school.
a) How far is the school from Kyle’s home?
b) At what speed does Kyle cycle?
10. State whether each relation is linear or nonlinear. Explain how you know.
4
a) y 3x 6
b) x 1 2
2
y
0
x
5
10 15 20 25
Time (min)
c)
d)
7
9
3
4
5
6
11
13
15
17
y = 5x 2 + 6x - 4
x
1 2
3
4
5
6
y 3 0
5
12
21
32
11. Solve.
a) x + 5 = 12
b) 13 = 9 - x
c) 2x = 18
d) -3x = - 21
e) 2x - 5 = 15
f ) 4x - 6 = 8x + 2
12. a) If 3x - 2y = 14 and x 1.5, determine the value of y.
b) If 0.36x + 0.54y = 1.1 and y 0.7, determine the value of x.
13. a) Make a concept map that shows different strategies you could use
to graph 2x 4y 8.
b) Which strategy would you use? Explain why.
6
Getting Started
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Getting Started
APPLYING What You Know
YOU WILL NEED
Making Change
• grid paper
• ruler
Barb is withdrawing $100 from her bank account. She asks for the money
in $5 bills and $10 bills.
?
Which combinations of $5 bills and $10 bills equal $100?
A.
If the teller gives Barb four $10 bills, how many $5 bills does he give her?
B.
List four more combinations of $100. Record the combinations in a table.
Number of $5 Bills
Number of $10 Bills
4
C.
Let x represent the number of $5 bills, and let y represent the number
of $10 bills. Write an equation for combinations of these bills with
a total value of $100.
D.
Graph your equation for part C. Should you use a solid or broken line?
Explain.
E.
Describe how the number of $10 bills changes as the number
of $5 bills increases.
F.
Explain what the x-intercept and y-intercept represent on your graph.
G.
Which points on your graph are not possible combinations?
Explain why.
H.
Determine all the possible combinations of $5 bills and $10 bills
that equal $100.
NEL
Chapter 1
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Representing Linear Relations
YOU WILL NEED
GOAL
• grid paper
• ruler
• graphing calculator
Use tables, graphs, and equations to represent linear relations.
LEARN ABOUT the Math
Aiko’s cell-phone plan is shown here.
Aiko has a budget of $30 each month
for her cell phone.
?
EXAMPLE 1
Services
Cost
calls
20¢/min
text messages
15¢/message
How can Aiko show how many messages and calls she can make
each month for $30?
Representing a linear relation
Show the combinations of messages and calls that are possible each month
for $30.
Aiko’s Solution: Using a table
Text Messages
Number
of
Cost
Messages
($)
Calls
Number
of
Cost
Minutes
($)
0
0
150
30
30
20
3
135
27
30
40
6
120
24
30
:
200
:
30
0
:
0
As the number of text messages increases, the
number of minutes available for calls decreases.
Aiko can make choices based on the numbers in
the table. For example, if Aiko sends 40 text
messages, she can talk for 120 min.
8
Total
Cost
($)
1.1 Representing Linear Relations
30
I made a table to show how many
messages and calls are possible for $30.
I started with 0 messages and let the
number of messages increase by 20
each time. I calculated the cost of the
messages by multiplying the number in
the first column by $0.15. Then I
subtracted the cost of the messages
from $30 to determine the amount of
money that was left for calls. I calculated
the number of minutes for calls by
dividing this amount by $0.20.
40 text messages a month is about
1 per day. 120 min a month for calls is
about 4 min per day.
NEL
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1.1
Malcolm’s Solution: Using an equation and a graph
Let x represent the number of text messages per
month. Let y represent the number of minutes
of calls per month.
Aiko has a budget of $30 for text messages and
calls, so 0.15x 0.20y 30.
Number of minutes of calls
At the x-intercept, y 0.
0.15x 0.20(0) 30
30
x =
0.15
x 200
240
At the y-intercept, x 0.
0.15(0) 0.20y 30
30
y =
0.20
y 150
I used letters for the variables.
I wrote an equation based on Aiko’s
budget of $30. In my equation, x text
messages cost $0.15x and y minutes of
calls cost $0.20y.
I used my equation to calculate the
maximum number of text messages and
the maximum time for calls. To do this,
I determined the intercepts.
Number of Minutes of Calls
vs. Number of Text Messages
y
200
I drew a graph by plotting the x-intercept
and y-intercept, and joining them.
160
0.15x 0.20y 30
120
I used a broken line because x represents
whole numbers only in this equation.
80
40
0
x
40 80 120 160 200 240
Number of text messages
The point (40, 120) shows that if Aiko sends
40 text messages in a month, she has a maximum
of 120 min for calls to stay within her budget.
Aiko’s options for text messages and
calls are displayed as points on the
graph. Each point on the graph
represents an ordered pair (x, y), where
x is the number of text messages per
month and y is the number of minutes
of calls per month.
Reflecting
A.
How does the table show that the relationship between the number
of text messages and the number of minutes of calls is linear?
B.
How did Malcolm use his equation to draw a graph of Aiko’s choices?
C.
Which representation do you think Aiko would find more useful:
the table or the graph? Why?
NEL
Chapter 1
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APPLY the Math
EXAMPLE 2
Career Connection
Careers as diverse as sales
consultants, software
developers, and financial
analysts have roles in currency
exchange.
Representing a linear relation using
graphing technology
Patrick has saved $600 to buy British pounds and euros for a school trip
to Europe. On the day that he goes to buy the currency, one pound
costs $2 and one euro costs $1.50.
a) Create a table, an equation, and a graph to show how many pounds
and euros Patrick can buy.
b) Explain why the relationship between pounds and euros is linear.
c) Describe how Patrick can use each representation to decide how much
of each currency he can buy.
Brittany’s Solution
a) Let x represent the pounds that
Patrick buys. Let y represent
the euros that he buys.
2x 1.50y 600
1.50y 600 2x
1.50y
600
2x
=
1.50
1.50
1.50
2
bx
y = 400 - a
1.50
Tech
Support
For help using a TI-83/84
graphing calculator to enter
then graph relations and
use the Table Feature, see
Appendix B-1, B-2, and B-6.
If you are using a TI-nspire,
see Appendix B-37, B-38,
and B-42.
10
1.1 Representing Linear Relations
I chose letters for the variables.
x pounds cost $2x and y euros
cost $1.50y. Patrick has $600.
I wrote an equation based on the
cost of the currency. I rearranged
my equation into the form
y mx b so I could enter
it into a graphing calculator.
I graphed the equation using
these window settings because
I knew that the y-intercept would
be at 400 and the x-intercept
would be at 300.
NEL
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1.1
I set the decimal setting to two
decimal places because x and y
represent money. Then I created
a table of values.
b) Since the degree of the equation
is one and the graph is a straight
line, the relationship is linear.
The first differences in the table
are constant.
In the table, each increase
of 1 in the x-values results
in a decrease of about 1.33
in the y-values.
Tech
Support
For help creating a difference
table with a TI-83/84 graphing
calculator, see Appendix B-7.
If you are using a TI-nspire,
see Appendix B-43.
c) By tracing up and down the line, or by scrolling up and down the table,
Patrick can see the combinations of pounds and euros. He can use the
equation, in either form, to calculate specific numbers of pounds or euros.
EXAMPLE 3
Selecting a representation for a linear
relation
Judy is considering two sales positions. Sam’s store offers $1600/month
plus 2.5% commission on sales. Carol’s store offers $1000/month plus
5% commission on sales. In the past, Judy has had about $15 000 in sales
each month.
a) Represent Sam’s offer so that Judy can check what her monthly pay
would be.
b) Represent the two offers so that Judy can compare them. Which offer
pays more?
Justine’s Solution
a) Let x represent her sales in dollars.
Let y represent her earnings
in dollars.
An equation will help Judy check
her pay.
y 1600 0.025x
NEL
I chose letters for the variables.
I wrote an equation to describe
what Judy’s monthly pay would
be. Her base salary is $1600.
Her earnings for her monthly
sales would be $0.025x,
2.5
since 2.5%
or 0.025.
100
Chapter 1
11
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Tech
Support
For help changing the
window settings and tracing
on a graph using a TI-83/84
graphing calculator, see
Appendix B-4 and B-2.
If you are using a TI-nspire,
see Appendix B-40 and B-38.
5/7/09
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Page 12
b) The equation for Carol’s offer is
y 1000 0.05x.
Judy can use a graph to compare
her pay for a typical month.
I wrote an equation for Carol’s
offer and graphed both relations
using a graphing calculator.
I adjusted the settings, as
shown, so I could see the point
where the graphs crossed.
I used Trace to compare the
two offers.
Sam’s offer pays more.
In Summary
Key Idea
• Three useful ways to represent a linear relation are
• a table of values
• a graph
• an equation
Need to Know
• A linear relation has the following characteristics:
• The first differences in a table of values are constant.
• The graph is a straight line.
• The equation has a degree of 1.
• The equation of a linear relation can be written in a variety of equivalent
forms, such as
• standard form: Ax By C 0
• slope y-intercept form: y mx b
• A graph and a table of values display some of the ordered pairs for a
relation. You can use the equation of a relation to calculate ordered pairs.
CHECK Your Understanding
1. Which of these ordered pairs are not points on the graph
of 2x 4y 20? Justify your decision.
a) (10, 0) b) (3, 7) c) (6, 2) d) (0, 5)
12
1.1 Representing Linear Relations
e) (12, 1)
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1.1
2. Jacob has $15 to buy muffins and doughnuts at the school bake sale,
as a treat for the Camera Club. Muffins are 75¢ each and doughnuts
are 25¢ each. How many muffins and doughnuts can he buy?
a) Create a table to show the possible combinations of muffins and
doughnuts.
b) What is the maximum number of muffins that Jacob can buy?
c) What is the maximum number of doughnuts that he can buy?
d) Write an equation that describes Jacob’s options.
e) Graph the possible combinations.
3. Refer to question 2. Which representation do you think is more useful
for Jacob? Justify your choice.
PRACTISING
4. State two ordered pairs that satisfy each linear relation and one ordered
K
pair that does not.
a) y 5x 1
b) 3x - 4y = 24
c) y 25x 10
d) 5x = 30 - 2y
5. Define suitable variables for each situation, and write an equation.
a) Caroline has a day job and an evening job. She works a total
of 40 h/week.
b) Caroline earns $15/h at her day job and $11/h at her evening job.
Last week, she earned $540.
c) Justin earns $500/week plus 6% commission selling cars.
d) Justin is offered a new job that would pay $800/week plus
4% commission.
e) A piggy bank contains $5.25 in nickels and dimes.
6. Graph the relations in question 5, parts a) and b).
7. Refer to question 5, parts c) and d). Justin usually has about $18 000
in weekly sales. Should he take the new job? Justify your decision.
8. Deb pays 10¢/min for cell-phone calls and 6¢/text message. She has
a budget of $25/month for both calls and text messages.
a) Create a table to show the ways that Deb can spend up to $25
each month on calls and text messages.
b) Create a graph to show the information in the table.
9. Leah earns $1200/month plus 3.5% commission.
C
NEL
a) Create an equation that she can use to check her paycheque
each month.
b) Last month, Leah had $96 174 in sales. Her pay before deductions
was $4566.09. Is this amount correct? Explain your answer.
Chapter 1
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10. Ben’s Bikes rents racing bikes for $25/day and mountain bikes
A
for $30/day. Yesterday’s rental charges were $3450.
a) Determine the greatest number of racing bikes that could have
been rented.
b) Determine the greatest number of mountain bikes that could have
been rented.
c) Write an equation and draw a graph to show the possible
combinations of racing and mountain bikes rented yesterday.
11. Abigail is planning to fly to Paris and then travel through Switzerland
and Austria to Italy by train. On the day that she goes to buy the
foreign currencies she needs, one euro costs $1.40 and one Swiss franc
costs $0.90. What combinations of these currencies can Abigail buy for
$630? Use two different strategies to show the possible combinations.
12. A student council invested some of the money from a fundraiser
in a savings account that pays 3%/year and the rest of the money
in a government bond that pays 4%/year. The investments earned
$150 in the first year.
a) Define two variables for the information, and write an equation.
b) Graph the information.
13. Maureen pays a $350 registration fee and an $85 monthly fee to belong
T
to a fitness club. Lia’s club has a higher registration fee but a lower
monthly fee. After five months, both Maureen and Lia have paid
$775. Determine the possible fees at Lia’s club.
14. a) Use the chart to show what you know about linear relations.
Characteristics:
Examples:
Methods of
Representation:
Linear Relation
Non-examples:
b) List the advantages and disadvantages of each of the three ways
to represent a linear relation. Describe situations in which each
representation might be preferred.
Extending
15. Create a situation that can be represented by each equation.
a) 0.10x + 0.25y = 4.65
b) y = 900 + 0.025x
16. Allan plans to create a new coffee blend using Brazilian beans that cost
$12/kg and Ethiopian beans that cost $17/kg. He is going to make
150 kg of the blend and sell it for $14/kg. Write and graph two
equations for this situation.
14
1.1 Representing Linear Relations
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Solving Linear Equations
GOAL
YOU WILL NEED
Connect the solution to a linear equation and the graph
of the corresponding relation.
• grid paper
• ruler
• graphing calculator
LEARN ABOUT the Math
Joe downloads music to his MP3 player from a site that charges $9.95 per
month plus $0.55 for each song. Joe has budgeted $40 per month to spend
on music downloads.
?
How can Joe determine the greatest number of songs
that he can download each month?
EXAMPLE 1
Selecting a strategy to solve the problem
Determine the maximum number of songs that Joe can download
each month.
William’s Solution: Solving a problem by reasoning
$40.00 $9.95 $30.05
I calculated how much of Joe’s
budget he can spend on the songs
he downloads, by subtracting the
$9.95 monthly fee from $40.
$30.05 , $0.55 54.63
Each song costs $0.55, so I
divided this into the amount he
would have left to spend on songs.
Joe can download a maximum
of 54 songs.
I rounded down to 54, since
55 songs would cost more than
he can spend.
Tony’s Solution: Solving a problem by using an equation
Let n represent the number of songs and let C represent the cost.
C 9.95 0.55n
40 9.95 0.55n
NEL
I created an equation and
substituted the $40 Joe has
budgeted for C.
Chapter 1
15
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40 9.95 9.95 0.55n 9.95
30.05 0.55n
30.05
= n
0.55
54.6 n
I solved for n using inverse
operations.
Joe can download a maximum
of 54 songs.
Since n has to be a whole
number, I used the nearest
whole number less than 54.6
for my answer.
Lucy’s Solution: Solving a problem using graphing technology
Let X represent the number of songs and Y1 the cost.
Y 1 9.95 0.55X
Tech
I entered the equation for the
cost of music downloads into
a graphing calculator. The
number of songs downloaded
has to be a whole number, so
X represents a whole number.
I graphed using Zoom Integer,
so the x-values would go up by
1 when I traced the graph.
I used Trace to determine which
point on the graph is closest
to y 40 (but less than $40).
This point is (54, 39.65).
Support
For help graphing and
tracing along relations using
a TI-83/84 graphing calculator,
see Appendix B-2. If you
are using a TI-nspire,
see Appendix B-38.
Joe can download 54 songs
in a month for $39.65.
Reflecting
16
A.
How are William’s and Tony’s solutions similar? How are they different?
B.
How did a single point on Lucy’s graph represent a solution
to the problem?
C.
Which strategy do you prefer? Explain why.
1.2 Solving Linear Equations
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1.2
APPLY the Math
EXAMPLE 2
Representing and solving a problem
that involves a linear equation
At 9:20 a.m., Adrian left Windsor with 64 L of gas in his car. He drove east
at 100 km/h. The low fuel warning light came on when 10 L of gas were
left. Adrian’s car uses gas at the rate of 8.8 L/100 km. When did the
warning light come on?
Stefani’s Solution: Solving an equation algebraically
Adrian’s car uses 8.8 L of gas
every 100 km. Since he drove
at 100 km/h, he used 8.8 L/h.
G 64 8.8t
10 64 8.8t
10 10 64 8.8t 10
0 54 8.8t
8.8t 54
54
t =
8.8
t 6.14
The warning light came on after
Adrian had been driving about 6.14 h.
0.14 60 8.4
The warning light came on about
6 h 8 min after 9:20 a.m., which
is about 3:28 p.m.
I calculated how much gas
the car used each hour.
I wrote an equation for the
amount of gas used. I let t
represent the time in hours, and
I let G represent the amount of
gas in litres.
The warning light came on
when G 10, so I let G 10
and solved for t using inverse
operations.
I wrote the time in hours and
minutes by multiplying the part
of the number to the right of
the decimal point by 60.
Henri’s Solution: Solving a problem by using a graph
y = 64 - 8.8x
NEL
I wrote an equation for the
amount of gas in the tank at any
time. I let x represent the time in
hours, and I let y represent the
amount of gas in litres.
Chapter 1
17
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Graph Y 1 = 64 - 8.8X.
I graphed the equation on a
graphing calculator. I knew that
the y-intercept was 64, and I
estimated that the x-intercept
was about 7, so I used the
window settings shown.
After about 6.17 h, there was
about 9.7 L of gas in the tank.
I used Trace to locate the point
with a y-value closest to 10.
Tech
Support
For help determining the point
of intersection between
two relations on a TI-83/84
graphing calculator, see
Appendix B-11. If you are
using a TI-nspire, see
Appendix B-47.
Based on the graph, the warning
light came on about 6.14 h after
Adrian started, at about 3:28 p.m.
To get an exact solution,
I entered the line Y2 10.
The x-coordinate of the point
of intersection between the
two lines tells the time when
10 L of gas is left in the tank.
In Summary
Key Idea
• You can solve a problem that involves a linear relation by solving
the associated linear equation.
Cost of Car Rental
vs. Distance
y
Cost ($)
100
Need to Know
• You can solve a linear equation in one variable by graphing the
associated linear relation and using the appropriate coordinate of
an ordered pair on the line. For example, to solve 3x 2 89, graph
y 3x 2 and look for the value of x at the point where y 89
on the line.
50
CHECK Your Understanding
0
18
x
1. Estimate solutions to the following questions using the graph at the left.
200 400 600 800
Distance (km)
1.2 Solving Linear Equations
a) What is the rental cost to drive 500 km?
b) How far can you drive for $80, $100, and $75?
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1.2
2. a) Write an equation for the linear relation in question 1.
b) Use your equation to answer question 1.
c) Compare your answers for question 1 with your answers for part b)
above. Which strategy gave the more accurate answers?
3. Apple juice is leaking from a carton at the rate of 5 mL/min.
There are 1890 mL of juice in the container at 10:00 a.m.
a) Write an equation for this situation, and draw a graph.
b) When will 1 L of juice be left in the carton?
PRACTISING
4. The graph at the right shows how the charge for a banquet hall
relates to the number of people attending a banquet.
a) Locate the point (160, 5700) on the graph. What do these
coordinates tell you about the charge for the banquet hall?
b) What is the charge for the banquet hall if 200 people attend?
c) Write an equation for this linear relation.
d) Use your equation to determine how many people can attend
for $3100, $4400, and $5000.
e) Why is a broken line used for this graph?
Cost of Banquet
vs. Number of People
y
9000
8000
7000
Cost ($)
K
5. Max read on the Internet that 1 U.S. gallon is approximately
equal to 3.785 L.
a) Draw a graph that you can use to convert U.S. gallons
into litres.
b) Use your graph to estimate the number of litres in 6 gallons.
c) Use your graph to estimate the number of gallons in 14 L.
6. Melanie drove at 100 km/h from Ajax to Ottawa. She left Ajax
6000
5000
4000
3000
2000
1000
0
x
50 100 150 200 250 300
Number of people
at 2:15 p.m., with 35 L of gas in the tank. The low fuel warning
light came on when 9 L was left in the tank. If Melanie’s SUV
uses gas at the rate of 9.5 L/100 km, estimate when the warning light
came on.
7. Hank sells furniture and earns $280/week plus 4% commission.
a) Determine the sales that Hank needs to make to meet his weekly
budget requirement of $900.
b) Write an equation for this situation, and use it to verify your
answer for part a).
8. The Perfect Paving Company charges $10 per square foot to install
A
NEL
interlocking paving stones, as well as a $40 delivery fee.
a) Determine the greatest area that Andrew can pave for $3500.
b) Andrew needs to include 5 cubic yards of sand, costing $15 per
cubic yard, to the total cost of the project. How much will this
added cost reduce the area that he can pave with his $3500 budget?
Chapter 1
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9. A student athletic council raised $4000 for new sports equipment and
uniforms, which will be purchased 3 years from now. Until then,
the money will be invested in a simple interest savings account that
pays 3.5%/year.
a) Write an equation and draw a graph to represent the relationship
between time (in years) and the total value of their investment.
b) Use the graph to determine the value of their investment after 2 years.
c) Use the equation to determine when their investment is worth
$4385.
10. Maria has budgeted $90 to take her grandmother for a drive. Katey’s
Kars rents cars for $65 per day plus $0.12/km. Determine how far
Maria and her grandmother can travel, including the return trip.
11. Cam earns $400/week plus 2.5% commission. He has been offered
C
another job that pays $700/week but no commission.
a) Describe three strategies that you could use to compare Cam’s
earnings for the two jobs.
b) Which job should Cam take? Justify your decision.
12. At 9:00 a.m., Chantelle starts jogging north at 6 km/h from the south
T
end of a 21 km trail. At the same time, Amit begins cycling south
at 15 km/h from the north end of the same trail. Use a graph
to determine when they will meet.
13. Explain how to determine the value of x, both graphically and
algebraically, in the linear relation 2x - 3y = 6 when y = 5.
Extending
14. The owner of a dart-throwing stand at a carnival pays 75¢ every time
the bull’s-eye is hit, but charges 25¢ every time it is missed. After
25 tries, Luke paid $5.25. How many times did he hit the bull’s-eye?
Health Connection
15. Adriana earns 5% commission on her sales up to $25 000, 5.5% on
Jogging is an exercise that
keeps you healthy and can burn
about 650 calories per hour.
Number of
Buttons
Cost per
Button ($)
1 to 25
1.00
26 to 50
0.80
51 to 100
0.60
101 or more
0.20
20
any sales between $25 000 and $35 000, 6% on any sales between
$35 000 and $45 000, and 7% for any sales over $45 000. Draw
a graph to represent how Adriana’s earnings depend on her sales.
What sales volume does she need to earn $2000?
16. A fabric store sells fancy buttons for the prices in the table at the left.
1.2 Solving Linear Equations
a) Make a table of values and draw a graph to show the cost
of 0 to 125 buttons.
b) Compare the cost of 100 buttons with the cost of 101 buttons.
What advice would you give someone who needed 100 buttons?
Comment on this pricing structure.
c) Write equations to describe the relationship between the cost and
the number of buttons purchased.
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Graphically Solving Linear
Systems
GOAL
Use graphs to solve a pair of linear equations simultaneously.
INVESTIGATE the Math
YOU WILL NEED
• grid paper
• ruler
• graphing calculator
Matt’s health-food store sells roasted almonds for $15/kg and dried
cranberries for $10/kg.
?
How can he mix the almonds and the cranberries to create
100 kg of a mixture that he can sell for $12/kg?
A.
Let x represent the mass of the almonds. Let y represent the mass
of the cranberries.
i) Write an equation for the total mass of the mixture.
ii) Write an equation for the total value of the mixture.
B.
Graph your equation of the total mass for part A. What do the points
on the line represent?
C.
Graph your equation of the total value for part A on the same axes.
What do the points on this line represent?
D.
Identify the coordinates of the point where the two lines intersect.
State what each value represents. How accurately can you estimate
these values from your graph?
E.
The equations for part A form a system of linear equations.
Explain why the coordinates for part D give the solution to
a system of linear equations.
F.
Substitute the coordinates into each equation to verify your solution.
Reflecting
G.
Explain why you needed two linear relations to describe the problem.
H.
Explain how graphing both relations on the same axes helped you
solve the problem.
I.
Explain why the coordinates of the point of intersection provide
an ordered pair that satisfies both relations.
NEL
system of linear equations
a set of two or more linear
equations with two or more
variables
For example, x y 10
4x 2y 22
solution to a system of linear
equations
the values of the variables in
the system that satisfy all the
equations
For example, (7, 3) is the
solution to
x y 10
4x 2y 22
Chapter 1
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APPLY the Math
EXAMPLE 1
Selecting a graphing strategy to solve a linear system
Solve the system y = 2x + 1 and x + 2y = - 8 using a graph.
Leslie’s Solution
The slope of
the line is 2.
2
rise
=
run
1
y = 2x + 1
At the y-intercept,
x 0.
y 2(0) 1
y1
I determined the slope and the
y-intercept of the first equation.
x + 2y = - 8
At the x-intercept, At the y-intercept,
y = 0.
x = 0.
x + 2(0) = - 8
0 + 2y = - 8
2y
x = -8
8
= 2
2
y = -4
y
0
-4
4
2y
8
I graphed the second line (red)
by plotting points (8, 0) and
(0,4) and joining them with a
straight line.
y
2
-2
2
2x
4
-2
x
I graphed the first line (blue) by
plotting the y-intercept and
using the rise and run to plot
another point on the line.
1
6
-8 -6 -4
I determined the x- and
y-intercepts of the second
equation.
x
-6
At the point of intersection, x = - 2
and y = - 3.
The solution is (2, 3).
y 2x 1
Left Side
y
3
22
Right Side
2x 1
2(2) 1
3
I located the point of intersection
and read its coordinates using
the axes of my graph.
x 2y 8
Left Side
x 2y
2 2(3)
8
1.3 Graphically Solving Linear Systems
Right Side
8
I checked my solution by
substituting the x- and y-values
into each equation.
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1.3
EXAMPLE 2
Solving a problem using a graphing strategy
Ellen drives 450 km from her university in Kitchener-Waterloo to her
home in Smiths Falls. She travels along one highway to Kingston at
100 km/h and then along another highway to Smiths Falls at 80 km/h.
The journey takes her 4 h 45 min. What is the distance from Kingston
to Smiths Falls?
Bob’s Solution
Let x represent the distance that
Ellen travels at 100 km/h. Let y
represent the distance that she
travels at 80 km/h.
I used letters to identify the variables
in this situation.
The total trip is 450 km, so
x y 450.
I wrote an equation for the total
distance.
y
x
3
+
= 4
100
80
4
x + y = 450
At the x-intercept, y 0. At the y-intercept, x 0.
0 y 450
x 0 450
y 450
x 450
y
3
x
+
= 4
100
80
4
At the x-intercept, y 0. At the y-intercept, x 0.
3
x
y
3
+ 0 = 4
= 4
0 +
100
4
80
4
3
3
x = 100 a 4 b
y = 80 a4 b
4
4
19
19
x = 100 a b
y = 80a b
4
4
x = 475
y = 380
NEL
distance
, then
time
distance
time =
.
speed
I wrote an equation to describe the total
x
time (in hours) for her trip, where
100
is the time spent driving at 100 km/h
y
and
is the time spent driving
80
at 80 km/h.
Since speed =
I determined the x- and y-intercepts
of the first equation.
I determined the x- and y-intercepts
of the second equation.
Chapter 1
23
Distance at 80 km/h (km)
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Distance at 80 km/h vs.
Distance at 100 km/h
y
I graphed each equation by plotting the
x- and y-intercepts and joining them
with a straight line.
500
400
300
200
100
0
x
100 200 300 400 500
Distance at 100 km/h (km)
I determined the coordinates of the point
of intersection. The y-coordinate of the
point is the distance.
The point of intersection is (350, 100), so
the distance from Kingston to Smiths Falls
is 100 km.
EXAMPLE 3
Selecting graphing technology to solve
a system of linear equations
Hayley wants to rent a car for a weekend trip. Kelly’s Kars charges $95 for
the weekend plus $0.15/km. Rick’s Rentals charges $50 for the weekend
plus $0.26/km. Which company charges less?
Elly’s Solution
24
Let x represent the distance driven
in kilometres. Let y represent the
total cost of the car rental in dollars.
I chose x and y for the variables.
The cost to rent a car from Kelly’s
Kars is y 0.15x 95.
I wrote an equation for the cost
to rent a car from Kelly’s Kars.
$0.15x represents the distance
charge and $95 represents the
weekend fee.
The cost to rent a car from Rick’s
Rentals is y 0.26x 50.
I wrote an equation for the cost
to rent a car from Rick’s Rentals.
$0.26x represents the distance
charge and $50 represents the
weekend fee.
1.3 Graphically Solving Linear Systems
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1.3
I used a graphing calculator. I
entered the equation for Kelly’s
Kars in Y1 and the equation for
Rick’s Rentals in Y2. I used a
thick line for Rick’s Rentals to
distinguish between the two lines.
I graphed the lines and adjusted
the window settings so that I
could see both lines and the
point of intersection.
The point of intersection is
(409, 156), to the nearest
whole numbers.
If Hayley drives 409 km, both
companies charge about $156.
If she plans to drive less than
409 km, Rick’s Rentals charges less.
If she plans to drive more than
409 km, Kelly’s Kars charges less.
The point of intersection
occurred when 0 … X … 600
and 0 … Y … 200. I used the
Intersect operation to determine
the point of intersection.
Tech
Support
To graph with a thick line
using a TI-83/84 graphing
calculator, scroll to the left of
Y2 and press ENTER .
Tech
Support
For help using a TI-83/84
graphing calculator to
determine the point
of intersection, see
Appendix B-11. If you
are using a TI-nspire,
see Appendix B-47.
I looked at my graph to
determine which line is lower
before and after the point
of intersection.
Before the point of intersection,
the thick line is lower, so Rick’s
Rentals charges less. After the
point of intersection, the thin line is
lower, so Kelly’s Kars charges less.
In Summary
Key Idea
• You can solve a system of linear equations by graphing both equations
on the same axes. The ordered pair (x, y) at the point of intersection
gives the solution to the system.
Need to Know
• You may not be able to determine an accurate solution to a system
of equations using a hand-drawn graph.
• To determine an accurate solution to a system of equations, you can
use graphing technology. When you use a graphing calculator, express
the equations in slope y-intercept form.
NEL
Chapter 1
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CHECK Your Understanding
1. Decide whether each ordered pair is a solution to the given system
of equations.
a) (2, 1); 3x 2y 4 and x 3y 5
b) (1, 4); x y 5 and 2x 2y 8
c) (1, 2); y 3x 5 and y 2x 4
d) (10, 5); x y 5 and y 5x 40
2. For each graph:
i) Identify the point of intersection.
ii) Verify your answer by substituting into the equations.
y
3x a)
4
y
x
2
14
y
2
x
-2
0
2
3y 4
1
x
-2
2
x
21
-2
0
2
-2
4
6
y
5
30
-4
x
4
6
8
-6
-8
2x
8
1
-2
0
y
y
2
-4
6
2
4
c)
y
-2
y
x
8
x
-4
b)
PRACTISING
3. a) Graph the system x y 5 and 3x 4y 12 by hand.
K
b) Solve the system using your graph.
c) Verify your solution using graphing technology.
4. Alex needs to rent a minivan for a week to take his band on tour.
Easyvans charges $230 plus $0.10/km. Cars for All Seasons charges
$150 plus $0.22/km.
a) Write an equation for each rental company.
b) Graph your equations.
c) Which rental company would you recommend to Alex? Explain.
5. Solve each linear system by graphing.
a) x y 3
xy7
b) x y 8
4x 2y 8
c) y 2x – 4
3x y 6
26
1.3 Graphically Solving Linear Systems
d) 2x y 10
yx2
e) y 3x – 5
y 2x 5
f ) 6x 5y 12 0
2x 5y 2 0
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1.3
6. Austin is creating a new “trail mix” by combining two of his best-selling
7. At Jessica’s Java, a new blend of coffee is featured each week. This
week, Jessica is creating a low-caffeine espresso blend from Brazilian
and Ethiopian beans. She wants to make 200 kg of this blend and sell
it for $15/kg. The Brazilian beans sell for $12/kg, and the Ethiopian
beans sell for $17/kg. How many kilograms of each kind of bean must
Jessica use to make 200 kg of her new blend of the week?
Mass of banana mix (kg)
blends: a pineapple–coconut–macadamia mix that sells for $18/kg and
a banana–papaya–peanut mix that sells for $10/kg. He is making 80 kg
of the new mix and will sell it for $12.50/kg. Austin uses the graph shown
at the right to determine how much of each blend he needs to use.
a) Write the equations of the linear relations in the graph.
b) From the graph, how much of each blend will Austin use?
Trail Mix
100
y
80
60
40
20
0
x
20 40 60 80 100
Mass of pineapple mix (kg)
8. When Arthur goes fishing, he drives 393 km from his home in Ottawa
to a lodge near Temagami. He travels at an average speed of 70 km/h
along the highway to North Bay and then at 50 km/h on the narrow
road from North Bay to Temagami. The journey takes him 6 h.
a) Write two equations to describe this situation.
b) Graph your equations.
c) Use your graph to determine the distance from North Bay
to Temagami.
9. Joanna is considering two job offers. Phoenix Fashions offers
$1500/month plus 2.5% commission. Styles by Rebecca offers
$1250/month plus 5.5% commission.
a) Create a linear system by writing an equation for each salary.
b) What value of sales would result in the same total salary
for both jobs?
c) Which job should Joanna take? Explain your answer.
10. Create a situation you can represent by a system of linear equations
T
that has the ordered pair (10, 15) as its solution.
11. Six cups of coffee and a dozen muffins originally cost $15.35. The price
of a cup of coffee increases by 10%. The price of a dozen muffins increases
by 12%. The new cost of six cups of coffee and a dozen muffins is $17.06.
Determine the new price of one cup of coffee and a dozen muffins.
12. Willow bought 3 m of denim fabric and 5 m of cotton fabric. The
total bill, excluding tax, was $22. Jared bought 6 m of denim fabric
and 2 m of cotton fabric at the same store for $28.
a) Write a linear system you can solve to determine the price
of denim fabric and the price of cotton fabric.
b) Solve your system using a graph.
c) How much will 8 m of denim fabric and 5 m of cotton fabric cost?
NEL
Chapter 1
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13. The drama department of a school sold 679 tickets to the school play, for
a total of $3370. Students paid $4 for a ticket, and non-students paid $7.
a) Write a linear system for this situation.
b) How many non-students attended the play? Solve the problem
by graphing your system.
14. The equations y 2, y 4x 2, and y 2x 10 form the sides
A
of a triangle.
a) Graph the triangle, and determine the coordinates of the vertices.
b) Calculate the area of the triangle.
15. A regular light bulb costs $0.65 to buy, plus $0.004/h for the
C
Environment Connection
Fluorescent light bulbs
decrease energy use and
reduce pollution levels.
electricity to make it work. A fluorescent light bulb costs $3.99 to buy,
plus $0.001/h for the electricity.
a) Write a cost equation for each type of light bulb.
b) Graph the system of equations using a graphing calculator. Use
the window settings 0 … X … 2000 and 0 … Y … 10.
c) After how long is the fluorescent light bulb cheaper than
the regular light bulb?
d) Determine the difference in cost after one year of constant use.
16. Rearrange the following sentences to describe the correct sequence of
steps for solving a problem by graphing a linear system. Discard any
sentences that do not belong in the description. Add any sentences
that are needed to make the description clearer.
• Label the graph.
• Verify the solution by substituting into both equations.
• Write two equations that describe the situation in the problem.
• Determine the slope of each line by calculating the rise over the run.
• Read the problem, and determine what you need to find.
• Graph both equations on the same set of axes.
• Choose the best strategy to graph each equation.
• Determine the coordinates of the point of intersection.
Extending
17. a) Solve the linear system 3x y 11 0 and x 2y 1 0.
b) Show that the line with the equation 9x 4y 19 0 passes
through the point where the lines in part a) intersect.
c) Determine the values of c and d if 9x 4y 19 0 is written
in the form c(3x y 11) d(x 2y 1) 0.
18. Solve the linear system y 2x 1, 4x 3y 7, and 6x y 17 0.
19. Solve each system of equations.
a) y = 2x 2
y = - 3x + 5
28
1.3 Graphically Solving Linear Systems
b) y = 2x
y = x - 1
NEL
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Curious Math
Curious Math
YOU WILL NEED
• ruler
Optical Illusions
An optical illusion occurs when an image you look at does not agree with
reality. The image has been deliberately created to deceive the eyes. Many
optical illusions are created using intersecting lines.
This image is called the Necker Cube. It is created
by drawing the frame of a cube. The intersecting
lines make the image confusing. When you focus
on the cube, it seems to move backward and
forward, causing you to question which is the
front and which is the back.
This image is called the Café Wall Illusion
because it was first noticed on café walls in
England. Dark and light “tiles” are arranged
alternately in staggered rows. Each row and
each tile is bordered by a shade of a colour that
is between the shades used for the “tiles.” This
has the strange effect of making the long
parallel lines appear crooked.
This image is called the Zöllner Illusion. The
long black lines appear not to be parallel, even
though they are. The short lines are drawn at an
angle to the long lines, giving the impression of
depth and tricking the eyes to perceive that one
end of each long line is closer to you than the
other end.
1. Do some research to find other examples of optical illusions that have
been created using intersecting lines.
2. Create your own optical illusion by drawing a series of intersecting lines.
NEL
Chapter 1
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
Q:
How can you represent the ordered pairs of a linear
relation?
A:
You can use variables to create an equation and then list the ordered
pairs in a table of values or plot them as points to create a graph.
• See Lesson 1.1,
Examples 1 to 3.
• Try Mid-Chapter Review
Questions 1 and 2.
EXAMPLE
Prices for cherries and peaches in July of one year are listed at the left.
What amounts of cherries and peaches can you buy for $15?
cherries
10.98
Solution
peaches
2.18
Write an equation to describe the relation:
• c kilograms of cherries cost $10.98c.
• p kilograms of peaches cost $2.18p.
• The total amount of money to buy the cherries and peaches is $15.
The equation 10.98c + 2.18p = 15 describes the linear relation between
p and c. Use this equation to calculate the approximate coordinates
of ordered pairs.
Amount of peaches (kg)
Fruit
Price per
Kilogram ($)
8
Amount of Cherries
vs. Peaches
y
6
Amount of Cherries, c (kg)
0
0.40
0.80
1.00
4
Amount of Peaches, p (kg)
15.00
= 6.88
2.18
4.87
2.85
1.84
2
x
0
0.4 0.8 1.2 1.6
Amount of cherries (kg)
Study
Aid
Graph the linear relation by hand, by plotting two or more ordered pairs
and drawing a straight line through the points.
Q:
How can you solve a linear equation in one variable
using a linear relation?
A:
Each ordered pair, or point, on the graph of a linear relation
represents the solution to the related linear equation.
• See Lesson 1.2, Examples 1
and 2.
• Try Mid-Chapter Review
Questions 3 to 5.
EXAMPLE
Consider 0.03x + 0.04y = 120, where x represents the amount of money
invested at 3%/year and y represents the amount invested at 4%/year.
The total interest earned for one year was $120. If $1500 was invested
at 3%/year, how much was invested at 4%/year?
Solution
Solve 0.03(1500) 0.04y 120.
30
Mid-Chapter Review
NEL
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Mid-Chapter Review
Solving the Equation Algebraically
Solving the Equation Graphically
0.03(1500) 0.04y 120
45 0.04y 120
45 0.04y 45 120 45
0.04y 75
0.04y
75
=
0.04
0.04
y 1875
Determine the point with an
x-coordinate of 1500 on the graph
of the relation. The y-coordinate
of this point is the solution
to the equation.
y = 1875
The amount invested at
4%/year was $1875.
The amount invested at
4%/year was $1875.
3000
y
0.03x 0.04y 120
2000
(1500, 1875)
1000
0
x
2000
4000
6000
Q:
How can you solve a linear system of equations
using graphs?
Study
A:
Graph the equations on the same axes by hand or using graphing
technology. The coordinates of the point where the lines intersect give
the solution to the system.
• Try Mid-Chapter Review
Aid
• See Lesson 1.3,
Examples 1 to 3.
Questions 6 to 10.
EXAMPLE
Solve y = x - 2 and 2x + 3y = 6.
Solution
Graphing by Hand
Graphing with Technology
Write the second equation in the
form y mx b:
3y 6 2x
y
3
2x 3y 6 2
2
3
1
x
-3
-2
-1
0
-1
-2
2
3
y 2 x or y x 2
1
2
3
4
5
Enter both equations, and use
the Intersect operation.
yx2
-3
From the graph, the solution is
x 2.5 and y 0.5.
NEL
From the graph on the calculator,
the solution is x 2.4 and y 0.4.
Chapter 1
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PRACTICE Questions
Lesson 1.1
5. Len plans to invest money he has saved so that
1. Doreen has $10 to buy apples and pears.
Fruit
Price per Kilogram ($)
apples
2.84
pears
2.18
Use each representation below to determine the
possible amounts of each type of fruit she can buy.
a) a table
b) a graph
c) an equation
2. Jon downloads music to his MP3 player from a
site that charges $12.95 per month and $0.45
per song. Another site charges $8.99 per month
and $0.95 per song. Compare the cost of the
two sites using a table and a graph.
Lesson 1.2
3. The graph shows how Sally’s weekly earnings
vary with the dollar value of the sales she makes
at a clothing store.
Weekly earnings ($)
Sally’s Earnings vs. Sales
500
400
(1500, 360)
200
100
0
x
1000
2000
Sales ($)
3000
4000
a) What do the coordinates (1500, 360) mean?
b) Use the graph to determine what Sally earns
when her sales are $3200.
c) Use the graph to determine what sales Sally
needs to make if she wants to earn $450.
d) Check your answers for parts b) and c)
algebraically.
4. a) Determine an equation for the perimeter
of any rectangle whose width is 8 cm less
than its length.
b) Determine the length of the rectangle whose
width is 72 cm.
32
Mid-Chapter Review
Lesson 1.3
6. Solve each linear system.
a) y = x + 4
y = - 2x + 1
b) y = 4x - 7
2x - 3y = 6
c) 3x - y = 3
2x + y = 2
d) 5x - 2y = 10
2x + 4y = 4
7. Solve each linear system.
y
300
he can earn $100 interest in one year. He will
deposit some of his money in an account that
pays 4%/year. He will use the rest of his money
to buy a one-year GIC that pays 5%/year.
a) Write an equation for Len’s situation, and
draw a graph.
b) Suppose that Len buys a GIC for $1500.
Use your graph to determine how much
he would need to put in the account.
c) Suppose that Len deposits $2200 in the
account. Determine how much he would
need for the GIC.
a) y = 5x - 8
10x - 5y = 7
c) x - 4y = - 1
- 3x + 8y = - 2
b) 2x + y = 2
1
x - y = 4
2
d) x + y = 0.7
5x - 4y = - 1
8. The art department at a school sold 323 tickets
to an art show, for a total of $790. Students paid
$2 for tickets, and non-students paid $3.50. The
principal asked how many non-students
attended the art show.
a) Write a system of two linear equations
for this situation.
b) Solve the problem by graphing the system.
9. Suppose you are solving the system y = 2x + m
and 3x - y = n, where m and n are integers.
Could this system have solutions in all four
quadrants? Justify your answer.
10. Create a situation that can be represented by a
system of linear equations that has the ordered
pair (5, 12) as its solution.
NEL
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Solving Linear Systems:
Substitution
GOAL
YOU WILL NEED
Solve a system of linear equations using an algebraic strategy.
• grid paper
• ruler
• graphing calculator
LEARN ABOUT the Math
Marla and Nancy played in a volleyball marathon for charity. They played
for 38 h and raised $412. Marla was sponsored for $10/h. Nancy was
sponsored for $12/h.
?
How many hours did each student play?
EXAMPLE 1
Selecting an algebraic strategy to solve
a linear system
Determine the length of time that each student played algebraically.
Isabel’s Solution
Let m represent the hours Marla
played. Let n represent the hours
Nancy played.
I used variables for the number
of hours each student played.
m + n = 38 10m + 12n = 412 I wrote one equation for the hours
played and another equation for
the money raised. I am looking for
an ordered pair (m, n) that satisfies
both equations.
m = 38 - n
10(38 - n) + 12n = 412
NEL
I decided to solve for a variable in
equation and then substitute
into equation . I solved for m
in equation since this was
easier than solving for n.
I used a substitution strategy
by substituting the expression for
m into equation . This gave
me an equation in one variable,
which included both pieces of
information I had.
substitution strategy
a method in which a variable in
one expression is replaced with
an equivalent expression from
another expression, when the
value of the variable is the same
in both
Chapter 1
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10(38) - 10n + 12n = 412
380 + 2n = 412
2n = 412 - 380
2n = 32
32
n =
2
n = 16
I used the distributive property
to multiply.
I solved for n using inverse
operations.
I solved for m by substituting
the value of n into the expression
for m.
m = 38 - n
m = 38 - 16
m = 22
Marla played for 22 h and Nancy
played for 16 h.
22 h 16 h 38h
22 h @ $10/h $220
16 h @ $12/h $192
$220 $192 $412
They raised $412.
I verified my solution. The total
number of hours played by both
girls and the total amount they
raised matches the information
in the problem.
Reflecting
A.
When you substitute to solve a linear system, does it matter which
equation or which variable you start with? Explain.
B.
Why did Isabel need to do the second substitution after solving for n?
C.
What would you do differently if you substituted for n instead of m?
APPLY the Math
EXAMPLE 2
Solving a problem modelled by a linear system using substitution
Most gold jewellery is actually a mixture of gold and copper. A jeweller is
reworking a few pieces of old gold jewellery into a new necklace. Some
of the jewellery is 84% gold by mass, and the rest is 75% gold by mass.
The jeweller needs 15.00 g of 80% gold for the necklace. How much
of each alloy should he use?
Wesley’s Solution
Let x represent the mass of the 84%
alloy in grams. Let y represent the
mass of the 75% alloy in grams.
34
1.4 Solving Linear Systems: Substitution
I used variables for the mass of each alloy.
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1.4
x y 15.00 I wrote an equation to represent the total mass of both alloys.
0.84x 0.75y 0.80(15)
0.84x 0.75y 12.00 I wrote another equation to represent the amount of pure
gold in the necklace, with the percents as decimals.
I calculated 80% of 15.00 as 12.00. This means that the final
15.00 g of the 80% alloy must contain 12.00 g of pure gold.
y 15.00 x
I solved for y in equation .
0.84x + 0.75(15.00 - x) = 12.00
0.84x + 11.25 - 0.75x = 12.00
I substituted 15.00 x for y in equation . Then I used
the distributive property to multiply.
0.84x - 0.75x = 12.00 - 11.25
0.09x = 0.75
0.75
x =
0.09
x 8.33
y 15.00 x
y 15.00 8.33
y 6.67
I solved for x, the mass of the 84% alloy, to the nearest
hundredth of a gram.
I substituted the value of x into y = 15.00 x and
calculated the mass of the 75% alloy to the nearest
hundredth of a gram.
The jeweller should use about 8.33 g of the
84% alloy and about 6.67 g of the 75% alloy.
EXAMPLE 3
Connecting the solution to a linear system
to the break-even point
Sarah is starting a business in which she will hem pants. Her start-up cost,
to buy a sewing machine, is $1045. She will use about $0.50 in materials
to hem each pair of pants. She will charge $10 for each pair of pants.
How many pairs of pants does Sarah need to hem to break even?
Robin’s Solution
Let x represent the number of pairs
of pants that Sarah hems, let C
represent her total costs, and let R
represent her revenue.
Tip
A company makes a profit
when it has earned enough
revenue from sales to pay its
costs. The point at which
revenue and costs are equal is
the break-even point.
I chose letters for the variables
in this problem.
C 1045 0.50x
The cost of materials to hem x pairs
of pants is $0.50x. I added the
start-up cost to get the total cost.
R 10x
Sarah charges $10 for each pair
of pants. When she hems x pairs
of pants, her revenue is $10x.
NEL
Communication
Chapter 1
35
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At the break-even point, total costs
and total revenue are the same.
y 1045 0.50x y 10x
10x 1045 0.50x
Because costs and revenue are
the same at the break-even
point, I used y to represent both
dollar amounts.
I substituted the expression for y
in equation into equation .
10x 0.50x 1045
9.50x 1045
1045
x =
9.50
x 110
I used inverse operations to solve
for x.
y 10(110)
y 1100
I determined y by substituting the
value of x into equation .
Check:
C 1045 0.50(110)
C 1045 55
C 1100
I verified my solution. The
break-even point is 110 pairs
of pants since the cost and
revenue are both $1100.
R 10(110)
R 1100
Sarah will break even when she
has hemmed 110 pairs of pants.
EXAMPLE 4
Selecting a substitution strategy to solve
a linear system
Determine, without graphing, where the lines with equations
5x 2y 2 and 2x 3y 16 intersect.
Carmen’s Solution
5x + 2y = - 2
2x - 3y = - 16
2y = - 2 - 5x
2y
-2 - 5x
=
2
2
5
y = -1 - x 2
36
1.4 Solving Linear Systems: Substitution
I decided to isolate the variable y
in equation . This resulted in an
equivalent form of the equation,
which I called equation .
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1.4
2x - 3 a - 1 -
5
x b = - 16
2
15
x = - 16
2
15
x = - 16 - 3
2x +
2
15
x = - 19
2x +
2
15
2(2x) + 2 a x b = 2( - 19)
2
4x + 15x = - 38
19x = - 38
- 38
19x
=
19
19
x = -2
2x + 3 +
5
( - 2)
2
10
y = -1 +
2
y = -1 + 5
y = 4
The values of x and y must satisfy
both equations at the point of
intersection, so I substituted the
expression for y in equation into equation .
I used the distributive property to
multiply, and then I simplified.
I multiplied all the terms in the
equation by the lowest common
denominator of 2 to eliminate
the fractions. Then I used inverse
operations to solve for x.
y = -1 -
I substituted the value of x into
equation . Then I determined
the value of y.
The lines intersect at the point (–2, 4).
Check by graphing:
2x - 3y = - 16
–3y = –2x - 16
2
16
y = x +
3
3
I solved for y in equation to
get the equation in the form
y mx b.
Tech
I graphed equations and using a graphing calculator. I
used the Intersect operation to
verify the point of intersection.
The graph confirms that the lines
intersect at (2, 4).
NEL
Support
For help using a TI-83/84
graphing calculator to
determine the point of
intersection, see Appendix B-11.
If you are using a TI-nspire,
see Appendix B-47.
Chapter 1
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In Summary
Key Idea
• To determine the solution to a system of linear equations algebraically:
• Isolate one variable in one of the equations.
• Substitute the expression for this variable into the other equation.
• Solve the resulting linear equation.
• Substitute the solved value into one of the equations to determine
the value of the other variable.
Need to Know
• Substitution is a convenient strategy when one of the equations can
easily be rearranged to isolate a variable.
• Solving the equation created by substituting usually involves the
distributive property. It may involve operations with fractional
expressions.
• To verify a solution, you can use either of these strategies:
• Substitute the solved values into the equation that you did not use
when you substituted.
• Graph both linear relations on a graphing calculator, and determine
the point of intersection.
CHECK Your Understanding
1. For each equation, isolate the indicated variable.
1
x + y = 10, x
2
b) 4x y 3 0, x d) 2x y 12, y
a) 10x y 1, y
c)
2. To raise money for a local shelter, some Grade 10 students held
Cars $6
Vans $8
38
a car wash and charged the prices at the left. They washed 53 vehicles
and raised $382.
a) Write an equation to describe the number of vehicles washed.
b) Write an equation to describe the amount of money raised
in terms of the number of each type of vehicle.
c) Solve for one of the variables in your equation for part a).
d) Substitute your expression for part c) into the equation for
part b). Solve the new equation.
e) Substitute your answer for part d) into your equation for
part a). Solve for the other variable. How many of each type
of vehicle did the students wash?
1.4 Solving Linear Systems: Substitution
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1.4
PRACTISING
3. Decide which variable to isolate in one of the equations in each
system. Then substitute for this variable in the other equation, and
solve the system.
a) x + 3y = 5
b) 2x + y = 4
2x - 3y = - 17
3x - 16y = 6
4. Solve for the indicated variable.
a) 8a = 4 - b, b
b) 6r + 3s = 9, r
d) 0.3x - 0.3y = 1.8, x
e) 0.12x - 0.06y = 0.24, y
1
1
c) 3u + 7v = 21, v
f ) x + y = 5, x
3
2
5. Decide which variable to isolate. Then substitute for this variable, and
K solve the system.
a) y = x - 5, x + y = 9
d) 3x - 2y = 10, x + 3y = 7
b) x = y + 4, 3x + y = 16
e) 2x + y = 5, x - 3y = 13
c) x + 4y = 21, 4x - 16 = y f ) x + 2y = 0, x - y = - 4
6. Tom pays a one-time registration charge and regular monthly fees
to belong to a fitness club. After four months, he had paid $420.
After nine months, he had paid $795. Determine the registration
charge and the monthly fee.
7. A health-food company packs almond butter in jars. Some jars hold 250 g.
Other jars hold 500 g. On Tuesday, the company packed 186.5 kg of
almond butter in 511 jars. How many jars of each size did they pack?
8. The difference between two angles in a triangle is 11°. The sum of
the same two angles is 77°. Determine the measures of all three angles
in the triangle.
9. Solve each system. Check your answers.
a) x + 3y = 7 and 3x - 2y = - 12
b) 3 = 2a - b and 4a - 3b = 5
c) 7m + 2n = 21 and 10m + 4n = - 10
d) 6x - 2y + 1 = 0 and 3x - 5y + 7 = 0
e) 3c + 2d = - 24 and 2c + 5d = - 38
1
1
1
f ) x - 3y = and x - 9y = 5
4
2
3
10. Dan has saved $500. He wants to open a chequing account at
Save-A-Lot Trust or Maple Leaf Savings. Using the information
at the right, which financial institution charges less per month?
11. Wayne wants to use a few pieces of silver to make a bracelet. Some of the
jewellery is 80% silver, and the rest is 66% silver. Wayne needs 30.00 g
of 70% silver for the bracelet. How much of each alloy should he use?
NEL
chequing accounts
$10 per month
plus
$0.75 per cheque
chequing accounts
$7 per month
plus
$1.00 per cheque
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12. Sue is starting a lawn-cutting business. Her start-up cost to buy two
A
lawn mowers and an edge trimmer is $665. She has figured out that
she will use about $1 in gas for each lawn. If she charges $20 per lawn,
what will her break-even point be?
13. A woodworking shop makes tables and chairs. To make a chair, 8 min
is needed on the lathe and 8 min is needed on the sander. To make
a table, 8 min is needed on the lathe and 20 min is needed
on the sander. The lathe operator works 6 h/day, and the sander
operator works 7 h/day. How many chairs and tables can they make
in one day working at this capacity?
Safety Connection
14. James researched these nutrition facts:
• 1 g of soy milk has 0.005 g of carbohydrates and 0.030 g of protein.
• 1 g of vegetables has 0.14 g of carbohydrates and 0.030 g of protein.
Eye protection must be worn
when operating a lathe.
James wants his lunch to have 50.000 g of carbohydrates and 20.000 g
of protein. How many grams of soy milk and vegetables does he need?
15. Nicole has been offered a sales job at High Tech and a sales job
at Best Computers. Which offer should Nicole accept? Explain.
• High Tech: $500 per week plus 5% commission
• Best Computers: $400 per week plus 7.5% commission
16. Monique solved the system of equations 2x y 4 and y 4x 10
2x (4x 10) 4
2x 4x 10 4
2x 10 4
2x 14
x 7
y 4(7) 10
y 28
C
by substitution. Her solution is at the left.
a) What did she do incorrectly?
b) Write a correct solution. Explain your steps.
17. Jennifer has nickels, dimes, and quarters in her piggy bank. In total,
T
she has 49 coins, with a value of $5.20. If she has five more dimes than
all the nickels and quarters combined, how many of each type of coin
does she have?
18. Explain why you think the strategy that was presented in this
lesson is called substitution. Use the linear system x 4y 8 and
3x 16y 3 in your explanation.
Extending
19. Marko invested $300 000 in stocks, bonds, and a savings account
at the rates shown at the left. He invested four times as much in
stocks as he invested in the savings account. After one year, he earned
$35 600 in interest. How much money did he put into each type
of investment?
40
1.4 Solving Linear Systems: Substitution
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Equivalent Linear Systems
GOAL
YOU WILL NEED
Compare solutions for equivalent systems of linear equations.
• grid paper
• ruler
LEARN ABOUT the Math
14
Cody solved this system of linear equations by graphing.
12
x + 2y = 10
4x - y = - 14
10
4x y 14
8
He concluded that the solution to this linear system
is the ordered pair (2, 6).
?
y
What other systems of linear equations have
the same solution?
(–2, 6)
6
x
4
2y
2
10
x
-6 -4 -2
0
-2
2
4
6
8
10
-4
EXAMPLE 1
Connecting addition and subtraction with
the equations of a linear system
Add and subtract the two equations in the system that Cody solved.
Graph all four equations on the same axes.
Sean’s Solution
Add the equations.
x + 2y = 10
4x - y = - 14
5x + y = - 4
Subtract the equations.
x + 2y = 10
4x - y = - 14
- 3x + 3y = 24
NEL
If x 2y 10 and 4x y 14,
(x 2y) (4x y) 10 (14).
This is another way to add the
equations. You can collect
like terms by adding the x terms,
y terms, and constants.
If x 2y 10 and 4x y 14,
(x 2y) (4x y) 10 (14).
This is another way to subtract the
equations. You can collect like
terms by subtracting the x terms,
y terms, and constants.
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The x-intercept for 5x y 4
is at (0.8, 0), and the y-intercept
is at (0, 4). The x-intercept for
3x 3y 24 is at (8, 0), and
the y-intercept is at (0, 8).
5x y 4
x 2
I determined the intercepts for
each new equation.
y
4x y 14
16
12 3x 3y 24
y6
8
(–2, 6)
4
x
-12 -8 -4
0
-4
4
8
12 16 20
x 2y 10
-8
x + 2y = 10
4x - y = - 14
4x - y = - 14 -3x + 3y = 24
4x - y = - 14
x + 2y = 10
5x + y = -4 -3x + 3y = 24
x + 2y = 10
5x + y = - 4
equivalent systems of linear
equations
two or more systems of linear
equations that have the same
solution
I graphed the new equations
in green on the same axes. Both
of the new lines pass through
(2, 6). I graphed the lines
x 2 and y 6 in red, as well.
These two lines also intersect at
(2, 6). All four new lines pass
through the point of intersection
for the original two lines. The
solution did not change when I
added or subtracted the equations
in the original system of equations.
Any pair of the two original and
two new equations can be used
to form a linear system that has
the same solution, (2, 6).
5x + y = - 4
-3x + 3y = 24
These are all equivalent systems of
linear equations.
The system of linear equations x 2
and y 6 is equivalent to the other
systems, but written in a simpler form that
directly shows the solution.
EXAMPLE 2
Connecting multiplication by a constant
with the equations of a linear system
Multiply x + 2y = 10 by 4 and 4x - y = - 14 by - 2. Graph
the original and two new equations on the same axes.
Donovan’s Solution
4(x + 2y) = 4(10)
4x + 8y = 40
42
1.5 Equivalent Linear Systems
I multiplied both sides of the first
equation by 4.
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1.5
- 2(4x - y) = - 2( - 14)
- 8x + 2y = 28
I multiplied both sides of the
second equation by 2.
4x 8y 40 has intercepts at (10, 0)
and (0, 5). 8x 2y 28 has
intercepts at (–3.5, 0) and (0, 14).
I determined the intercepts
for the new equations.
x 2
y
16
I graphed the equations on the
same axes.
4x y 14
8x 2y 28
12
y6
8
4
x
-12 -8 -4
0
-4
-8
4 8 12 16 20
x 2y 10
4x 8y 40
x + 2y = 10
4x - y = - 14
x + 2y = 10
- 8x + 2y = 28
4x + 8y = 40
4x - y = - 14
4x + 8y = 40
- 8x + 2y = 28
The new lines were identical to
the original lines, and the point
of intersection was unchanged
at (2, 6). I graphed the lines
x 2 and y 6, as well.
These two lines also intersect at
(2, 6). When I multiplied each
equation by a constant, their
graphs did not change.
Multiples of either original
equation, combined together,
form a linear system that has
the same solution, (2, 6).
These are all equivalent systems
of linear equations.
The system of linear equations x 2
and y 6 is equivalent to the other
systems, but it is written in a simpler
form that directly shows the solution.
Reflecting
A.
NEL
Suppose that Sean had doubled one equation first before he added.
Would the new equation go through the same point of intersection
as the original equations did?
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B.
What would have happened to the graphs if the equations in Donovan’s
solution had been multiplied by constants other than 4 and 2?
C.
Why does solving a system of linear equations result in an equivalent
system, with a horizontal line and a vertical line?
APPLY the Math
EXAMPLE 3
Reasoning about equivalent linear systems
a) Solve the system 2x 5y 4 and x 2y 7.
b) Show that an equivalent system is formed by multiplying the second
equation by 2 and then adding and subtracting the equations.
Emma’s Solution
a) x - 2y = 7
x = 7 + 2y
I decided to solve the system
using substitution. I used the
second equation to isolate x.
Since x has a coefficient of 1 in
this equation, I was able to avoid
working with fractions.
2(7 + 2y) + 5y = - 4
I substituted 7 2y for x in the
first equation.
14 + 4y + 5y = - 4
14 + 9y - 14 = - 4 - 14
9y = - 18
y = -2
I solved the new equation for y.
x 7 2y
x 7 2(2)
x74
x3
Then I substituted 2 for y and
solved for x.
The solution is the ordered pair (3, –2).
b)
44
1.5 Equivalent Linear Systems
2(x - 2y) = 2(7)
2x - 2(2y) = 2(7)
2x - 4y = 14
I multiplied both sides of the
second equation by 2.
2x + 5y = - 4
2x - 4y = 14
4x + y = 10
I added the new equation to the
first equation in the system.
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1.5
2x + 5y = - 4
2x - 4y = 14
0x + 9y = - 18
Then I subtracted the new
equation from the first equation.
4x + y = 10
9y = - 18
I wrote the new system that
was formed by adding and
subtracting. The coefficient for
x was 0.
y = -2
I solved for y in 9y = - 18.
4x + y = 10
4x - 2 = 10
4x = 10 + 2
4x = 12
x = 3
I substituted the value of y
into 4x + y = 10 and
determined x.
The solution to the new system
is the ordered pair (3, 2).
Since the solutions are the same,
the new system 4x y 10
and 9y 18 is equivalent to
the original system.
In Summary
Key Ideas
• When you add and subtract the equations in a linear system, you create
an equivalent linear system of equations that has the same solution as
the original system.
• When you multiply one or both equations of a system by a constant
other than 0, you create an equivalent linear system of equations that
has the same solution as the original system.
Need to Know
• When you add and subtract the equations of a linear system, the graphs
of the new equations are different from the graphs of the original
equations. However, they pass through the same point of intersection.
• Multiplying an equation by a constant other than 0 does not change
the graph of the equation.
• Adding or subtracting the equations of a linear system may result
in a simpler equation, with only one variable.
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CHECK Your Understanding
1. Add and subtract the equations in each linear system to create
a new linear system.
a) x - 3y = 2
b) x - y = 2
2x + y = - 5
2x + 3y = 19
c) 3x + y = 3
x - 2y = 8
d) 4x + 2y = 8
-x - 2y = - 4
2. a) Solve each original system in question 1.
b) Verify that the original system and the new system for question 1
have the same solution.
PRACTISING
y
6
3. a) State the solution to the linear system that is shown in the graph
at the left.
b) Create a new linear system by adding and then subtracting
the equations of the original system.
c) Substitute the x- and y-values for part a) into each of the new
equations for part b) to verify that the new system and the original
system are equivalent.
4
2
-2
2y
0
-2
-4
2
2x
x
6 8
3y
5
6
x
4. a) Multiply one of the equations in the graph in question 3 by any
integer other than zero.
b) Determine the x- and y-intercepts of the new equation to verify
that the graph of the new equation is the same as the graph
of the original equation in question 3.
c) Repeat parts a) and b) for the other equation in question 3.
5. a) Use substitution to solve the linear system 4x + y = 1 and
x - 2y = - 11.
b) Create another linear system by adding and subtracting
the equations in part a).
c) Verify that the systems in parts a) and b) have the same solutions.
6. A linear system is defined by x + 2y = 2 and –2x - y = 5.
K
a) Multiply the first equation by 3 and the second equation by 2.
b) Create another linear system by adding and subtracting
the equations you formed for part a).
c) Use a graph to show that the systems in parts a) and b) are equivalent.
7. a) Use substitution to solve the linear system - 2x + 5y = 2 and
x - 3y = - 2.
b) Multiply the first equation by 3 and the second equation by 5.
c) Add and subtract the equations for part b) to create a new linear
system.
d) Use substitution to verify that the systems in parts a) and c)
are equivalent.
46
1.5 Equivalent Linear Systems
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1.5
8. a) You have seen that multiplying an equation by a non-zero constant
C
does not change the graph of the equation. How do you think
dividing an equation by a non-zero constant would affect its graph?
Explain your reasoning.
b) Use a graph to solve the linear system:
3x - 3y = 6
8x + 4y = 4
c) Divide the first equation by 3 and the second equation by 4. Then
graph the new system you created. What effect, if any, did dividing
the equations have on the graph?
d) Add and subtract the equations for part c) to create a new linear
system.
e) Determine whether the systems for parts b) and d) are equivalent.
9. a) Solve this linear system:
T
4x + y = 4
2x - 3y = - 5
b) Show that your solution also satisfies the equation 2x + 11y = 23.
c) Determine constants a and b so that a times the first equation
in the system added to b times the second equation results
in the equation in part b).
10. a) Create two linear systems that are equivalent to this system:
3x - 4y = 3
-x - y = 6
b) Verify that all three systems have the same solution.
11. a) Why might you solve this system of equations by adding or subtracting?
A
2x + 3y = 4
2x - 3y = 8
b) Create a system of linear equations that you might solve by adding.
c) Create a system of linear equations that you might solve
by subtracting.
12. Add and subtract the equations in this system.
Use the new equations to solve it.
2x + 3y = 4
2x - 3y = 8
13. The sum of two numbers is 33, and their difference is 57.
a) Create a system of linear equations for this situation.
b) Create an equivalent system by adding and subtracting
your equations.
c) Solve the equivalent system to determine the two numbers.
NEL
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14. As the owner of a banquet hall, you are in charge of catering a
reception. You are serving two dinners: a chicken dinner that costs $20
and a fish dinner that costs $18. Two hundred guests have ordered
their dinners in advance, and the total bill is $3880.
a) Create a system of linear equations for this situation.
b) Create an equivalent system by multiplying the guest equation by
20 and then subtracting the cost equation from this new equation.
c) Simplify and then solve the equivalent system to determine
the number of each type of dinner ordered.
Career Connection
Banquet halls offer careers in
financial management, cooking,
catering, and hospitality.
15. a) Candice claims that these systems of linear equations
are equivalent. Is she correct? Justify your decision.
System A
System B
System C
3x - 2y = 2
- 7x + y = 10
x = -2
-10x + 3y = 8
13x - 5y = - 6
y = -4
b) Create another system of linear equations that is equivalent
to the systems in part a).
16. a) What are equivalent systems of linear equations?
b) How can you use the equations for a linear system to create
an equivalent system?
c) How can this help you to solve the original system?
Extending
17. If you create equivalent linear systems in which there is only
one variable in one or more of the new equations, you can solve
the original system without graphing it. Use this strategy to solve
the following linear systems.
a) x - 4y = - 22
b) 3x - 4y = 30
2x + y = 1
2x + 5y = - 26
18. Consider this system of linear equations:
2x + y = 7
8x + 4y = 28
a) Can you create an equivalent system that contains only
one variable?
b) What does your result for part a) suggest about the solution
to the original system?
c) What does your result for part a) suggest about the graphs
of both lines?
19. Repeat question 18 for the system 2x + y = 7 and 8x + 4y = 10.
48
1.5 Equivalent Linear Systems
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Solving Linear Systems:
Elimination
GOAL
Solve a linear system of equations using equivalent equations
to remove a variable.
LEARN ABOUT the Math
Every day, Brenna bakes chocolate chip and low-fat oatmeal cookies in her
bakery. She uses different amounts of butter and oatmeal in each recipe.
Brenna has 47 kg of butter and 140 kg of oatmeal.
?
Chocolate Chip
Low-Fat Oatmeal
• 13 kg butter
• 2 kg butter
• 8 kg oatmeal
• 29 kg oatmeal
How many batches of chocolate chip and low-fat oatmeal
cookies can Brenna bake?
EXAMPLE 1
Selecting an algebraic strategy to eliminate
a variable
Determine the number of batches of each type of cookie that Brenna
can bake using all the butter and oatmeal she has.
Chantal’s Solution: Selecting an algebraic strategy
to eliminate a variable
Let r represent the number of batches
of chocolate chip cookies. Let s represent
the number of batches of low-fat cookies.
13r + 2s = 47
8r + 29s = 140
butter
oatmeal
8(13r + 2s ) = 8(47)
8(13r) + 8(2s ) = 8(47)
104r + 16s = 376
13(8r + 29s ) = 13(140)
13(8r) + 13(29s ) = 13(140)
104r + 377s = 1820
NEL
8
I used variables for the numbers
of batches.
I wrote two equations, to
represent the amount of butter
and the amount of oatmeal.
I decided to use an elimination
strategy to eliminate the r terms
by subtracting two equations.
To eliminate the r terms by
subtracting, I had to make the
coefficients of the r terms the
same in both equations. I
13
multiplied equation by 8 and
equation by 13.
elimination strategy
a method of removing a
variable from a system of linear
equations by creating an
equivalent system in which the
coefficients of one of the
variables are the same or
opposites
Communication
Tip
The steps that are required to
eliminate a variable can be
described by showing the
operation and the equation
number. For example,
“ 8” means “equation multiplied by 8.”
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104r + 16s =
376
104r + 377s = 1820
-361s = - 1444
I subtracted the equations to
eliminate r. 8 13
-1444
-361
s = 4
I solved for s.
s =
13r + 2(4) = 47
13r + 8 = 47
13r = 47 - 8
13r = 39
39
r =
13
r = 3
I substituted the value of s into
equation . (I could have used
equation instead, if I had
wanted.) I solved for r.
Brenna can make three batches of chocolate
chip cookies and four batches of low-fat cookies.
I verified my answers.
Check:
Type of
Cookie
Number
of Batches
Butter (kg)
Oatmeal (kg)
chocolate chip
3
3 13 39
3 8 24
low-fat
4
428
4 29 116
39 8 47
24 116 140
Total
Leif’s Solution: Selecting an algebraic strategy to eliminate
a different variable
13r + 2s = 47
8r + 29s = 140
29(13r + 2s) = 29(47) 29
29(13r) + 29(2s) = 29(47)
377r + 58s = 1363
-2(8r + 29s) = -2(140) 2
- 2(8r) - 2(29s) = -2(140)
-16r - 58s = - 280
50
1.6 Solving Linear Systems: Elimination
I started with the same linear
system as Chantal, but I decided
to eliminate the s terms by
adding the two equations.
To eliminate the s terms by adding,
I had to make the coefficients of
the s terms opposites. To do this,
I multiplied equation by 29 and
equation by 2.
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1.6
377r + 58s = 1363
- 16r - 58s = - 280
361r
= 1083
1083
361
r = 3
r =
13(3) + 2s = 47
39 + 2s = 47
2s = 47 - 39
2s = 8
8
s =
2
s = 4
I added the equations to
eliminate s. 29 2
I solved for r.
I substituted the value of r into
equation .
Brenna can make three batches of chocolate
chip cookies and four batches of low-fat cookies.
Reflecting
A.
How did Chantal and Leif use elimination strategies to change
a system of two equations into a single equation?
B.
Why did Chantal and Leif need to multiply both equations
to eliminate a variable?
C.
Explain when you would add and when you would subtract
to eliminate a variable.
D.
Whose strategy would you choose: Chantal’s or Leif ’s? Why?
APPLY the Math
EXAMPLE 2
Selecting an elimination strategy
to solve a linear system
Use elimination to solve this linear system:
7x - 12y = 42
17x + 8y = - 2
NEL
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John’s Solution
7x - 12y = 42
17x + 8y = - 2
I decided to eliminate the
y terms because I prefer to add.
Since their signs were different,
I could make the coefficients
of the y terms opposites.
14x - 24y = 84
51x + 24y = - 6
65x
= 78
78
x =
65
x = 1.2
2
3
The coefficients of y are factors
of 24. I multiplied equation by 2 and equation by 3 to
make the coefficients of the
y terms opposites. Then I added
the new equations to eliminate y.
23
7(1.2) - 12y = 42 8.4 - 12y = 42
- 12y = 42 - 8.4
- 12y = 33.6
33.6
y =
-12
y = - 2.8
I substituted the value of x into
equation and solved for y.
Verify by substituting x = 1.2 and
y = - 2.8 into both original equations.
7x 12y 42
Left Side
Right Side
7x 12y
42
7(1.2) 12(2.8)
8.4 33.6
42
17x 8y 2
Left Side
Right Side
17x 8y
2
17(1.2) 8(2.8)
20.4 22.4
2
The solution is (1.2, 2.8).
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1.6 Solving Linear Systems: Elimination
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1.6
EXAMPLE 3
Selecting an elimination strategy to solve a system
with rational coefficients
During a training exercise, a submarine travelled 20 km/h on
the surface and 10 km/h underwater. The submarine travelled
200 km in 12 h. How far did the submarine travel underwater?
Tanner’s Solution
Let x represent the distance that the submarine
travelled on the surface. Let y represent the
distance that it travelled underwater.
I wrote an equation for the total distance travelled
during the training exercise.
x + y = 200
The time spent on the surface is
The time spent underwater is
y
x
+
= 12
20
10
x
.
20
y
.
10
y
x
+
b = 20(12)
20
10
y
x
20 a b + 20 a b = 20(12)
20
10
x + 2y = 240
20 a
I used variables for the distances that the
submarine travelled.
20
I used the formula time distance
to write
speed
expressions for the time spent on the surface
and the time spent underwater. Then I wrote
an equation for the total time.
I created an equivalent system with no fractional
coefficients by multiplying equation by 20, since
20 is a common multiple of 20 and 10.
20
x + y = 200 x + 2y = 240 20
- y = - 40
y = 40
x + 40 = 200
x = 200 - 40
x = 160
Since the coefficients of x were now the same,
I decided to eliminate x by subtracting
the equations.
To determine x, I substituted 40 for y
in the equation x y 200.
The submarine travelled 40 km underwater.
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In Summary
Key Idea
• To eliminate a variable from a system of linear equations, you can
• add two equations when the coefficients of the variable are opposite
integers
• subtract two equations when the coefficients of the variable
are the same
Need to Know
• Elimination is a convenient strategy when the variable you want
to eliminate is on the same side in both equations.
• If there are fractional coefficients in a system of equations, you can
form equivalent equations without fractional coefficients by choosing
a multiplier that is a common multiple of the denominators.
• Adding, subtracting, multiplying, or dividing both sides of a linear
equation in the same way produces an equation that is equivalent
to the original equation.
CHECK Your Understanding
1. For each linear system, state whether you would add or subtract
to eliminate one of the variables without using multiplication.
a) 4x + y = 5 b) 3x - 2y = 8 c) 4x - 3y = 6 d) 4x - 5y = 4
3x + y = 7
5x - 2y = 9
4x + 7y = 9
3x + 5y = 10
2. a) Describe how you would eliminate the variable x from the system
of equations in question 1, part a).
b) Describe how you would eliminate the variable x from the system
of equations in question 1, part c).
3. When a welder works for 3 h and an apprentice works for 5 h,
they earn a total of $175. When the welder works for 7 h and
the apprentice works for 8 h, they earn a total of $346. Determine
the hourly rate for each worker.
PRACTISING
Safety Connection
Welders must wear a helmet
with a mask that has a
darkened lens, as well as
gloves and clothing that are
flame- and heat-resistant.
54
4. To eliminate y from each linear system, by what numbers
would you multiply equations and ?
a) 4x + 2y = 5 3x - 4y = 7 c) 4x + 3y = 12 -2x + 5y = 7 b) 3x - 7y = 11 5x + 8y = 9 d) 9x - 4y = 10 3x + 2y = 10 1.6 Solving Linear Systems: Elimination
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1.6
5. To eliminate x from each linear system in question 4, by what numbers
would you multiply equations and ?
6. Solve each system by using elimination.
K
a) 3x + y = - 2
x - y = -6
b) x + 5y = 1
2x + 3y = 9
c) 4x - y = 5
-5x + 2y = - 1
d) 2x - 3y = - 2
3x - y = 0.5
e) 3x - 2y = - 39
x + 3y = 31
f ) 5x - y = - 3.8
4x + 3y = 7.6
7. Determine, without graphing, the point of intersection for the lines
with equations x + 3y = - 1 and 4x - y = 22. Verify your solution.
8. In a charity walkathon, Lori and Nicholas walked 72.7 km.
Lori walked 8.9 km farther than Nicholas.
a) Create a linear system to model this situation.
b) Solve the system to determine how far each person walked.
9. The perimeter of a beach volleyball court is 54 m. The difference
between its length and its width is 9 m.
a) Create a linear system to model this situation.
b) Solve the system to determine the dimensions of the court.
10. Rolf needs 500 g of chocolate that is 86% cocoa for a truffle recipe.
A
He has one kind of chocolate that is 99% cocoa and another kind that
is 70% cocoa. How much of each kind of chocolate does he need to
make the 86% cocoa blend? Round your answer to the nearest gram.
11. Determine the point of intersection for each pair of lines. Verify
your solution.
a) 4x + 7y = 23
c) 0.5x - 0.3y = 1.5
6x - 5y = - 12
0.2x - 0.1y = 0.7
y
x
x
b)
= - 2 d)
- 5y = 7
11
8
2
y
y
x
23
= 3
3x + =
2
4
2
2
e) 5x - 12y = 1
13x + 9y = 16
y - 3
x
f)
+
= 1
9
3
x
- (y + 9) = 0
2
12. Each gram of a mandarin orange has 0.26 mg of vitamin C and
0.13 mg of vitamin A. Each gram of a tomato has 0.13 mg of vitamin C
and 0.42 mg of vitamin A. How many grams of mandarin oranges and
tomatoes have 13 mg of vitamin C and 20.7 mg of vitamin A?
13. On weekends, as part of his exercise routine, Carl goes for a run, partly
C
NEL
on paved trails and partly across rough terrain. He runs at 10 km/h on
the trails, but his speed is reduced to 5 km/h on the rough terrain. One
day, he ran 12 km in 1.5 h. How far did he run on the rough terrain?
Chapter 1
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17
. If the
12
3
numerators are switched, the sum is . Determine the two fractions.
2
14. Two fractions have denominators 3 and 4. Their sum is
15. A student athletic council raised $6500 in a volleyball marathon.
The students put some of the money in a savings account and the rest
in a government bond. The rates are shown at the left. After one year,
the students earned $235. How much did they invest at each rate?
3% simple interest
annually
4% simple interest
annually
16. The caterers for a Grade 10 semi-formal dinner and dance are
preparing two different meals: chicken at $12 or pasta at $8.
The total cost of the dinners for 240 students is $2100.
a) How many chicken dinners did the students order?
b) How many pasta dinners did they order?
17. A magic square is an array of numbers with the same sum across
T
any row, column, or main diagonal.
16
2
A
8
B
24
14
9
A
2
18
B
a) Determine a system of linear equations you can use to determine
the values of A and B in both squares.
b) What are the values of A and B?
18. Explain what it means to eliminate a variable from a linear system.
Use the linear system 3x + 7y = 31 and 5x - 8y = 91 to compare
different strategies for eliminating a variable.
Extending
19. The sum of the squares of two negative numbers is 74. The difference
of their squares is 24. Determine the two numbers.
20. Solve the system 2xy + 3 = 4y and 3xy + 2 = 5y .
21. A general system of linear equations is
ax + by = e
cx + dy = f
where a, b, c, d, e, and f are constant values.
a) Use elimination to solve for x and y in terms of a, b, c, d, e, and f.
b) Are there any values that a, b, c, d, e, and f cannot have?
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1.6 Solving Linear Systems: Elimination
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Exploring Linear Systems
YOU WILL NEED
GOAL
• graphing calculator, or
Connect the number of solutions to a linear system
with its equations and graphs.
grid paper and ruler
EXPLORE the Math
Three different linear systems are given below.
A
B
C
2x 3y 4
2y 6 3x
xy5
–4x 3y 1
6x 5 4y
3x 15 3y
?
How many solutions can a linear system have, and how can you
predict the number of solutions without solving the system?
A.
Solve each system of linear equations algebraically. Record the number
of solutions you determine.
Linear System
A
B
C
2x 3y 4
4x 3y 1
3x 2y 6
6x 4y 5
x y 5
3x 3y 15
Number of
Solutions
B.
Examine your algebraic solution for system A. How do you think
the lines that represent the equations in this system intersect? Explain.
Graph the system to check your conjecture.
C.
Repeat part B for each of the other two systems.
D.
For each system, explain how the graphical solution is related
to the algebraic solution, and vice versa.
E.
Examine the equations in each system. Are there clues that tell you
how the lines will intersect? Explain.
F.
Can a linear system of two equations have exactly two solutions?
Can it have exactly three solutions? Explain.
G.
Discuss the different cases you have identified and how each case
relates to the equations and their corresponding graphs.
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Reflecting
H.
Both equations in a linear system that has no solution are written
in the form Ax + By = C . Describe the relationship between
the coefficients and the constants. What does this tell you about
the graphs of both lines?
I.
Both equations in a linear system that has an infinite number
of solutions are written in the form Ax + By = C . Describe
the relationship between the coefficients and the constants. What
does this tell you about the graphs of both lines?
J.
How can you tell, by looking at the equations, that a linear system
has exactly one solution?
In Summary
Key Idea
• A linear system can have no solution, one solution, or an infinite number of solutions.
Need to Know
• When a linear system has no
solution, the graphs of both
lines are parallel and never
intersect. For example, the
system 3x 2y 8 and
6x 4y 40 does not have a
solution. The coefficients in
the equations are multiplied by
the same amount, but the
constants are not.
y
3x
2y
-2
2
4
6
-4
58
3x 2y 8
4
1.7 Exploring Linear Systems
6x 4y 16
3x 2y 8
4
2
x
x
8
0
6
2
2
-2
xy2
40
4
y
8
6
4y
6
• When a linear system has an
infinite number of solutions, the
graphs of both equations are
identical and intersect at every
point. For example, the system
3x 2y 8 and 6x 4y 16
has an infinite number of
solutions. The coefficients and
constants in the equations are
multiplied by the same amount.
y
8
6x
8
-6 -4
• When a linear system has
one solution, the graphs of the
two lines intersect at a single
point. For example, the system
3x 2y 8 and x y 2
has one solution. The
coefficients and constants
in the equations are not
multiplied by the same
amount.
-6 -4
-2
0
-2
-4
2
4
(4, –2)
6
x
-6 -4
-2
0
-2
2
4
6
-4
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1.7
FURTHER Your Understanding
1. Graph a linear system of equations that has each number of solutions.
a) none
b) one
c) infinitely many
2. Use the equation 3x + 4y = 2.
a) Write another equation that will create a linear system
with each number of solutions.
i) none
ii) one
iii) infinitely many
b) Verify your answers for part a) algebraically and graphically.
3. Predict the number of solutions for each linear system. Then test
your predictions by solving each system algebraically and verify
with graphing technology.
a) y = 3x - 5
e) 2x + 3y = 10
y = 4x + 6
10x + 15y = 50
b) y = 4x - 3
f ) 3x - 5y - 2 = 0
y = 4x - 7
4x + 5y + 2 = 0
3
c) y = 5x g) y = 1.25x - 0.375
2
5y = 4x
y = 5x - 1.5
d) x + 2y = 10
h) 2x - 5 = 4y
y = 8 - 0.5x
0.01x - 0.02y = 0.25
4. Create a system of linear equations that has each number of solutions.
Then verify the number of solutions algebraically and graphically.
a) none
b) one
c) infinitely many
5. Both equations in a linear system are written in the form
Ax + By = C . Explain how you could predict the number of
solutions using the coefficients and constants of the two equations.
6. An air traffic controller is plotting the course of two jets scheduled
to land in 15 min. One aircraft is following a path defined by
the equation 3x - 5y = 20 and the other by the equation
18x = 30y + 72. Should the controller alter the paths of either
aircraft? Justify your decision.
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Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
• See Lesson 1.4,
Q:
How can you use algebra to solve a linear system?
A1:
You can use a substitution strategy. Express one variable in one of the
equations in terms of the other variable. Substitute this expression
into the other equation, and solve for the remaining variable. Finally,
substitute the solved value into the expression to determine the value
of the other variable.
Examples 1 to 4.
• Try Chapter Review
Questions 7 to 9.
EXAMPLE
Solve the system.
2x + y = 29
4x - 3y = 18
Solution
From , y = 29 - 2x.
Substitute this expression for y into and solve for x.
4x - 3(29 - 2x) = 18
4x - 87 + 6x = 18
10x - 87 = 18
10x = 18 + 87
10x = 105
x = 10.5
Determine y by substituting this value of x into the expression for y.
y = 29 - 2(10.5)
y = 8
The solution is x = 10.5 and y = 8.
Study
Aid
• See Lesson 1.6,
Examples 1 and 3.
• Try Chapter Review
Questions 12 to 16.
A2:
You can use an elimination strategy. Eliminate x or y from the system
by multiplying one or both equations by a constant other than zero,
and then adding or subtracting the equations. Solve the resulting
equation for the remaining variable. Substitute the solved value into
one of the original equations, and determine the value of the other
variable.
EXAMPLE
60
Chapter Review
Solve the system.
2x + y = 29
4x - 3y = 18
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Chapter Review
Solution
Eliminating x
Eliminating y
Multiply by 2 and subtract.
4x + 2y = 58 2
4x - 3y = 18
5y = 40
y = 8
Substitute y = 8 into .
2x + 8 = 29
2x = 29 - 8
2x = 21
x = 10.5
Multiply by 3 and add.
3
6x + 3y = 87
4x - 3y = 18
10x = 105
x = 10.5
Substitute x = 10.5 into .
2(10.5) + y = 29
21 + y = 29
y = 29 - 21
y = 8
The solution is x = 10.5 and y = 8.
Q:
When you use elimination, how do you decide whether
to add or subtract the equations?
A:
If the coefficients of the variable you want to eliminate are the same,
subtract. If the coefficients are opposites, add.
Study
Aid
• See Lesson 1.6,
Examples 1 to 3.
• Try Chapter Review
Questions 12 and 15.
Q:
How do you decide whether to use graphs, substitution,
or elimination to solve a linear system?
A:
The strategy you use will depend on what degree of accuracy is
required, what form the equations are written in, and whether more
than just the solution is required. Algebraic solutions give exact
answers, whereas hand-drawn graphs often do not.
Use graphs if
• you do not need an exact answer
• you need to look for trends or compare the graphs before and after
the point of intersection
• you have a graphing calculator and both equations are in the form
y = mx + b
Use substitution if
• you need exact answers
• one of the variables in the equation is already isolated, ready
to make the substitution (that is, in the form y = mx + b)
• you can easily rearrange one equation to isolate a variable
Use elimination if
• you need exact answers
• both equations are in the form Ax + By = C or Ax + By + C = 0
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PRACTICE Questions
Lesson 1.1
4. Readycars charges $59/day plus $0.14/km
1. Sheila is planning to visit relatives in England
and Spain. On the day that she wants to buy the
currencies for her trip, one euro costs $1.50 and
one British pound costs $2.00. What
combinations of these currencies can Sheila buy
for $700? Use three different strategies to show
the possible combinations.
2. After a fundraiser, the treasurer for a minor
soccer league invested some of the money in a
savings account that paid 2.5%/year and the rest
in a government bond that paid 3.5%/year.
After one year, the money earned $140 in
interest. Define two variables, write an equation,
and draw a graph for this information.
Lesson 1.2
3. Gary drove his pickup truck from Cornwall to
Chatham. He left Cornwall at 8:15 a.m. and
drove at a steady 100 km/h along Highway 401.
The graph below shows how the fuel in the tank
varied over time.
to rent a car. Bestcars charges $69/day plus
$0.11/km. Describe three different strategies
you could use to compare these two rental rates.
What advice would you give someone who
wants to rent a car from one of these companies?
Lesson 1.3
5. Solve each linear system graphically.
a) x + y = 2
x = 2y + 2
b) y - x = 1
2x - y = 1
6. Tools-R-Us rents snowblowers for a base fee
of $20 plus $8/h. Randy’s Rentals rents
snowblowers for a base fee of $12 plus $10/h.
a) Create an equation that represents the cost
of renting a snowblower from Tools-R-Us.
b) Create the corresponding equation
for Randy’s Rentals.
c) Solve the system of equations graphically.
d) What does the point of intersection mean
in this situation?
Amount of gasoline (L)
Amount of Fuel vs. Time
y
60
50
40
30
(5, 25.5)
20
Lesson 1.4
10
0
x
2
4 6 8
Time (h)
10
a) What do the coordinates of the point
(5, 25.5) tell you about the amount of fuel?
b) How much fuel was in the tank at
11:45 a.m.?
c) The low fuel warning light came on when
6 L of fuel remained. At what time did this
light come on?
62
Chapter Review
7. Use substitution to solve each system.
a) 2x + 3y = 7
-2x - 1 = y
b) 3x - 4y = 5
x - y = 5
c) 5x + 2y = 18
2x + 3y = 16
d) 9 = 6x - 3y
4x - 3y = 5
8. Courtney paid a one-time registration fee to join
a fitness club. She also pays a monthly fee. After
three months, she had paid $315. After seven
months, she had paid $535. Determine the
registration fee and the monthly fee.
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Chapter Review
9. A rectangle has a perimeter of 40 m. Its length
is 2 m greater than its width.
a) Represent this situation with a linear system.
b) Solve the linear system using substitution.
c) What do the numbers in the solution
represent? Explain.
Lesson 1.5
13. Lyle needs 200 g of chocolate that is 86% cocoa
for a cake recipe. He has one kind of chocolate
that is 99% cocoa and another kind that is 70%
cocoa. How much of each kind of chocolate
does he need to make the cake? Round your
answer to the nearest gram.
14. A Grade 10 class is raising money for a school-
10. a) Which linear system below is equivalent
to the system that is shown in the graph?
y
6
y
2x
4
2
4x
8
-4
-2
0
-2
2
5y
2
building project in Uganda. To buy 35 desks
and 3 chalkboards, the students need to raise
$2082. To buy 40 desks and 2 chalkboards, they
need to raise $2238. Determine the cost of
a desk and the cost of a chalkboard.
x
4
6
-4
-6
-8
A. 2x - 5y = 4
B. x - 3y = - 1
-x + y = 1
2x + y = 4
b) Use addition and subtraction to create
another linear system that is equivalent
to the system in the graph.
c) Use multiplication to create another linear
system that is equivalent to the system
in the graph.
11. a) Create two linear systems that are equivalent
to the following system.
-2x - 3y = 5
3x - y = 9
b) Verify that all three systems have the same
solution.
Lesson 1.6
12. Use elimination to solve each linear system.
a) 2x - 3y = 13
5x - y = 13
b) x - 3y = 0
3x - 2y = - 7
NEL
c) 3x + 21 = 5y
4y + 6 = - 9x
1
d) x - y = - 1
3
2
1
x - y = -1
3
4
15. Solve the linear system.
2(2x - 1) - (y - 4) = 11
3(1 - x) - 2(y - 3) = - 7
16. Juan is a cashier at a variety store. He has a total
of $580 in bills. He has 76 bills, consisting of
$5 bills and $10 bills. How many of each type
does he have?
17. a) Sketch a linear system that has no solution.
b) Determine two possible equations that
could represent both lines in your sketch.
c) Explain how the slopes of these lines
are related.
18. The linear system 6x + 5y = 10 and
ax + 2y = b has an infinite number
of solutions. Determine a and b.
Chapter 1
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Chapter Self-Test
1. The Rainin’ Raisins company packs raisins in 500 g cartons and 750 g
Number
of 500 g
Cartons
Number
of 750 g
Bags
50
1150
100
1117
bags. One day, 887.5 kg of raisins were packed into full cartons and
bags. Ben wondered how many cartons and how many bags could have
been packed. He began to chart some possible combinations shown
at the left. Use a graph and an equation to show all the possible
combinations.
150
1083
2. Milk is leaking from a carton at a rate of 4 mL/min. There is 1500 mL
200
1050
:
:
:
:
1000
517
1500
183
:
:
of milk in the carton at 8:30 a.m.
a) Write an equation and draw a graph for this situation.
b) Determine graphically when 1 L of milk will be left in the carton.
c) Use your equation to determine algebraically when 1 L of milk
will be left in the carton.
3. Solve each system of equations. Use a different strategy for each system.
a) 3x + y = - 2
5x - y = - 10
b) 2x + 7 = y
-3x - 2y = 10
c) 3x + 5y = - 9
2x - y = 7
4. Shannon needs 20 g of 80% gold to make a pendant. She has some
Process
Checklist
✔ Questions 1 and 2: Did you
connect and compare the
representations?
✔ Questions 5 and 9: Did you
communicate your
thinking clearly?
✔ Questions 7 and 8: Did you
problem solve by selecting
and applying appropriate
strategies?
85% gold and some 70% gold, from broken jewellery, and wants
to know how much of each she should use. Determine the quantity
of each alloy that she should use.
5. How would you explain to someone why it makes sense that you can
add two equations and subtract them to create an equivalent system
of linear equations?
6. a) Create two linear systems that are equivalent to the system
3x + 4y = - 8 and x - 2y = 9.
b) Verify that all three systems are equivalent.
7. In his spare time, Kim likes to go cycling. He cycles partly on paved
surfaces and partly off-road, through hilly and wooded areas. He cycles
at 25 km/h on paved surfaces and at 10 km/h off-road. One day, he
cycled 41 km in 2 h. How far did he cycle off-road?
8. Last summer, Betty earned $4200 by painting houses. She invested
some of the money in a savings account that paid 3.5%/year and the
rest in a government bond that paid 4.5%/year. After one year, she has
earned $174 in interest. How much did she invest at each rate?
9. Explain how you know that this system of equations has no solution.
15 - 6y = 9x
3x + 2y = 8
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Chapter Self-Test
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Chapter Task
Coefficient Clues
Each morning, Stuart adds 250 g of fruit to his yogurt as a source of
vitamin C. Today, he wants to use one of the following combinations
of two fruits:
• pear and pineapple
• banana and blueberry
• apple and cranberry
Fruit
Amount of Vitamin C
in 1 g of Fruit (mg)
apple
0.15
banana
0.10
blueberry
0.10
cherry
0.10
cranberry
0.15
grapefruit
0.40
kiwi
0.70
mango
0.53
orange
0.49
Health Connection
pear
0.04
pineapple
0.25
Researchers claim that vitamin C
can help prevent colds, heart
disease, and cancer.
strawberry
0.60
?
How can Stuart determine which 250 g fruit combinations
will provide 25 mg of vitamin C?
A.
Write a system of linear equations to model each of today’s possible
fruit combinations.
B.
Solve each system.
C.
How many solutions does each system have?
D.
Examine at least three other fruit combinations using the data in the
table. Repeat parts A to C to determine if any of these combinations
will provide 25 mg of vitamin C.
E.
NEL
Describe the 250 g fruit combinations that will provide 25 mg
of vitamin C.
Task
Checklist
✔ Did you label all your
graphs?
✔ Did you answer all the
questions completely?
✔ Did you check your
answers?
✔ Did you explain your
thinking clearly?
Chapter 1
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Chapter
2
Analytic
Geometry: Line
Segments and
Circles
GOALS
You will be able to
• Use coordinates to determine and solve
problems involving midpoints, slopes,
and lengths of line segments
• Determine the equation of a circle
with centre (0, 0)
• Use properties of line segments to
identify geometric figures and verify
their properties
? Architects often design buildings
and structures that contain
arches. Carpenters may use
wooden frames to build these
arches. To build a wooden frame,
carpenters need to know the
radius of the circle that contains
the arch.
How can you determine the
radius of an arch like the ones
in these structures?
NEL
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Getting Started
WORDS YOU NEED to Know
1. Match each word with the diagram that best represents it.
a) diagonal
b) scalene triangle
c) perpendicular bisector
d) median of a triangle
i)
iii)
v)
ii)
B iv)
vi)
e) midsegment of a triangle
f ) Cartesian coordinate system
g) isosceles triangle
h) equilateral triangle
vii)
(–3, 1)
viii)
y
4
(2, 3)
2
x
(0, 0)
-4
-2
(–1.5, –2.5)
-4
A
Study
-2
0
2
4
SKILLS AND CONCEPTS You Need
Aid
• For more help and practice,
see Appendix A-4.
c2
The Pythagorean Theorem
The hypotenuse is the longest side in a right triangle. If c represents the
hypotenuse, and a and b represent the other two sides, a 2 + b 2 = c 2.
EXAMPLE
a2
a c
b
A right triangle has two perpendicular sides that measure 15 cm and 8 cm.
Calculate the length of the hypotenuse.
Solution
b2
Draw a right triangle with a = 8,
b = 15, and c as the hypotenuse.
b 15 cm
a 8 cm
c?
a2 + b2 = c 2
82 + 152 = c 2
289 = c 2
2289 = c
17 = c
The hypotenuse is 17 cm long.
2. Calculate each indicated side length. Round to the nearest tenth,
if necessary.
a)
x
b)
5m
x
200 mm
55 mm
12 m
68
Getting Started
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Getting Started
The Slope and the Equation of a Line
run x2 x1
y
rise y2 y1
0
P
R
The equation of a line may be written in
Q
4
(x2, y2)
x
x
-4
(x1, y1)
V
The slope of a line is the
rise divided by the run:
rise
m =
run
y2 - y1
m =
x2 - x1
y
8
0
-4
-8
4
8
T
U
• slope y-intercept form: y = mx + b,
where m is the slope and b is the
y-intercept
• standard form: Ax + By + C = 0
Parallel lines have the same slope:
m PT = m RU
Perpendicular lines have slopes
that are negative reciprocals:
1
1
; m QV = m QV = m PT
m RU
EXAMPLE
a) Determine the equation of the line through A( -1, 7) and B(2, 6).
b) Show that line AB is perpendicular to the line y = 3x - 2.
Solution
a) 햲 Determine the slope of AB.
햳 Determine the y-intercept of AB.
y2 - y1
1
m AB =
The equation is y = - x + b; (x, y) = (2, 6).
x2 - x1
3
6 - 7
1
=
6 = - (2) + b
2 - ( - 1)
3
1
2
= 6 + = b
3
3
햴 Write the equation.
20
= b
1
20
3
The equation is y = - x +
.
3
3
b) The line y = 3x - 2 has slope 3.
1
The negative reciprocal of 3 is - , the slope of line AB. So, line AB,
3
1
20
defined by y = - x +
, is perpendicular to the line y = 3x - 2.
3
3
3. Determine the equation of the line that
a) passes through points (-5, 3) and (7, 7)
1
b) is perpendicular to y = x + 7 and passes through ( -1, -2)
4
c) is parallel to y = - 5x + 6 and passes through (4, -3)
NEL
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PRACTICE
Aid
Study
5/7/09
• For help, see the Review
of Essential Skills and
Knowledge Appendix.
Question
Appendix
4
A-2, A-8
5a) to c)
A-9
4. Simplify each expression.
3
3
y - y
4
8
2 3
3
e) a b a b +
3 5
4
1
3
( -6) +
2
2
3
3
b)
8
7
2
x + 11x
c)
3
a)
d)
f ) ( -1.5)(0.625) + (4)( -0.125)
5. Solve.
4
(2x + 1) = 49
3
1
1
1
(x + 3) + (x - 2) = b)
4
3
2
x + 4
x - 2
= 1
c)
4
3
a) 3(7 - 4x) -
d) x 2 = 36
e) x 2 + 16 = 25
f ) 225 + x 2 = 289
6. Determine the point of intersection for each pair of lines.
a) y = 2x + 5
y = 3x + 4
b) 4x + 2y = 7
6x - 4y = 0
7. Determine the mean for each set of numbers.
a) 7, - 11, 23, 5
1 2
b) - ,
6 3
8. a) Calculate the area of the
shaded region. Round to
one decimal place.
c) -1.4, 3.6, - 0.1
b) Calculate the area and
perimeter of the shaded region.
Round to one decimal place.
11.6 cm
0
4.8 cm
12.1 cm
4.8 cm
9. Draw a Venn diagram to show relationships for the following figures.
quadrilateral
square
rectangle
70
Getting Started
trapezoid
parallelogram
rhombus
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Getting Started
APPLYING What You Know
YOU WILL NEED
• grid paper, ruler, and
protractor, or dynamic
geometry software
Diagonal Patterns
Jamie was creating a picture for her art portfolio using combinations of
intersecting line segments. She noticed that whenever she joined the
endpoints of a pair of these line segments, she created a quadrilateral.
?
How can you predict the type of quadrilateral by using the
properties of its diagonals?
A.
On a grid, plot four points and join them to create a square.
B.
Verify that your figure is a square by measuring its sides and
interior angles.
C.
Draw the diagonals in your figure, and measure
i) their lengths
ii) the angles formed at their point of intersection
Tech
D.
What do you notice about the diagonals you drew for part C?
E.
Repeat parts A to D for several more squares to determine whether
your observations are the same.
F.
Copy the table, and record your observations for parts A to E.
Type of
Quadrilateral
Side
Relationships
Interior
Angle
Relationships
Diagonal
Relationships
Support
For help using dynamic
geometry software to plot
points, construct lines, and
measure lengths and angles,
see Appendix B-18, B-21,
B-29, and B-26.
Relationship of
Angles Formed
by Intersecting
Diagonals
Diagram
square
rectangle
parallelogram
rhombus
isosceles trapezoid
kite
G.
Repeat parts A to E for each of the other figures in the table. Record
your observations.
H.
Explain how you could use what you learned to help you distinguish
between the figures in each pair.
i) a square and a rhombus
iii) a rhombus and a parallelogram
ii) a square and a rectangle
iv) a rhombus and a kite
NEL
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Midpoint of a Line Segment
YOU WILL NEED
GOAL
• grid paper, ruler, and
Develop and use the formula for the midpoint of a line segment.
compass, or dynamic
geometry software
INVESTIGATE the Math
Ken’s circular patio design for a client is
shown at the left. He is planning the layout
on a grid. He starts by drawing a circle that
is centred at the origin. Then he marks
points A, B, and C on the circumference
of the circle to divide it into thirds. He joins
these points to point O, at the centre
of the circle. He needs to draw semicircles
on the three radii: OA, OB, and OC.
y
A(x, y)
y
4
B
A
2
x
-6 -4
-2
0O
-2
2
4
6
-4
-6 C
?
How can Ken determine the coordinates of the centre
of the semicircle he needs to draw on radius OA?
A.
Construct a line segment like OA on a coordinate grid, with O
at (0, 0) and A at a grid point. Name the coordinates of A(x, y).
B.
Draw right triangle OAD, with side OD on the x-axis and side OA
as the hypotenuse.
C.
Draw a vertical line from E, the midpoint of OD, to M, the midpoint
of OA. Explain why ^OME is similar to ^OAD. Explain how the sides
of the triangles are related. Estimate the coordinates of M.
D.
Record the coordinates of point M. Explain why this is the centre
of the semicircle that Ken needs to draw.
x
O
6
D
Reflecting
72
E.
Why does it make sense that the coordinates of point M are the means
of the coordinates of points O and A?
F.
Suppose that point O had not been at (0, 0) but at another point
instead. If (x 1, y1) and (x 2, y2) are endpoints of a line segment, what
formula can you write to represent the coordinates of the midpoint?
Why does your formula make sense?
2.1 Midpoint of a Line Segment
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2.1
APPLY the Math
EXAMPLE 1
Reasoning about the midpoint formula
when one endpoint is not the origin
Determine the midpoint of a line segment with endpoints A(10, 2)
and B(6, 8).
Robin’s Solution: Using translations
8
y
B
I drew AB by plotting points A
and B on a grid and joining them.
6
4
2
B run 4
0
2 4
-2
-4
-6
A
6
8
x
10 12
rise –6
To make it easier to calculate the
midpoint of AB, I decided to
translate AB so that one endpoint
would be at the origin. I moved
point B to the origin by
translating it 6 units left and
8 units down. I did the same
to point A to get (4, -6) for A¿ .
A
-8
I could see that the run of A¿B¿
was 4 and the rise was - 6.
B¿(6 - 6, 8 - 8) = B¿(0, 0)
A¿(10 - 6, 2 - 8) = A¿(4, - 6)
x-coordinate of midpoint M¿
4
= 0 +
2
= 2
I determined the x-coordinate of
the midpoint of A¿B¿ by adding
half the run to the x-coordinate
of B¿ .
y-coordinate of midpoint M¿
-6
= 0 +
2
= -3
The midpoint of line
segment A¿B¿ is (2, - 3).
I determined the y-coordinate of
the midpoint of A¿B¿ by adding
half the rise to the y-coordinate
of B¿ .
M AB = M(2 + 6, - 3 + 8)
M AB = (8, 5)
The midpoint of line segment AB
is (8, 5).
NEL
To determine the coordinates of M,
the midpoint of AB, I had to undo
my translation. I added 6 to the
x-coordinate of the midpoint and
8 to the y-coordinate.
Chapter 2
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Sarah’s Solution: Calculating using a formula
(x, y) = a
x 1 + x 2 y1 + y2
,
b
2
2
x 1 = 10, y1 = 2
x 2 = 6, y2 = 8
10 + 6 2 + 8
,
b
2
2
16 10
= a , b
2 2
= (8, 5)
(x, y) = a
I decided to use the midpoint
formula.
I chose point A(10, 2) to be
(x1, y1) and point B(6, 8) to
be (x2, y2).
I substituted these values into
the midpoint formula.
The midpoint of line segment AB is (8, 5).
Tech
Support
I verified my calculations by
constructing AB using dynamic
geometry software. Then I
constructed the midpoint and
measured the coordinates of all
three points. My calculations
were correct.
For help constructing and
labelling a line segment,
displaying coordinates, and
constructing the midpoint
using dynamic geometry
software, see Appendix B-21,
B-22, B-20, and B-30.
EXAMPLE 2
Reasoning to determine an endpoint
1
1
Line segment EF has an endpoint at E a2 , -3 b . Its midpoint is
8
4
1
1
located at M a , - 1 b . Determine the coordinates of endpoint F.
2
2
Ali’s Solution
F
-2
-1
y
1
x
0
-1
1 2 3
1
1
, –1—
)
M (—
2
2
-2
rise
-3
74
run
4
I reasoned that if I could calculate the run and rise
between E and M, adding these values to the
x- and y-coordinates of M would give me the
x- and y-coordinates of F.
1
1
, –3 —
)
E (2—
8
4
2.1 Midpoint of a Line Segment
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2.1
Run = x 2 - x 1
1
1
= 2 8
2
4
17
=
8
8
13
5
=
or 1
8
8
F
-2
-1
Rise = y2 - y1
1
1
= - 3 - a -1 b
4
2
6
13
+
= 4
4
7
3
= - or -1
4
4
I let M = (x1, y1) and E = (x2, y2). Then I calculated
the rise and the run.
y
1
x
0
1
-1
2
3
To get to F, I had to start at M and move
3
5
1 units left and 1 units up.
8
4
4
M
-2
-3
E
x-coordinate of F
5
1
- 1
=
2
8
13
4
=
8
8
9
1
= - or - 1
8
8
y-coordinate of F
3
1
= -1 + 1
2
4
6
7
= - +
4
4
1
=
4
5
I subtracted 1 from the x-coordinate of M and
8
3
added 1 to the y-coordinate.
4
1 1
The coordinates of F are a - 1 , b .
8 4
EXAMPLE 3
Connecting the midpoint to an equation
of a line
A triangle has vertices at A(- 3, -1), B(3, 5), and C(7, - 3). Determine
an equation for the median from vertex A.
Graeme’s Solution
y
6
I plotted A, B, and C and joined
them to create a triangle.
B
4
2
-6 -4
A
-2
-2
-4
NEL
M
0
2
4
6
x
8
C
I saw that the side opposite vertex
A is BC, so I estimated the location
of the midpoint of BC. I called this
point M. Then I drew the median
from vertex A by drawing a
straight line from point A to M.
Chapter 2
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x 1 + x 2 y1 + y2
,
b
2
2
3 + 7 5 + (-3)
,
b
M BC = a
2
2
= (5, 1)
M = a
y2 - y1
x2 - x1
1 - (-1)
m AM =
5 - (- 3)
2
=
8
1
=
4
Slope =
To determine the equation of AM,
I had to calculate its slope. I used
the coordinates of A as (x1, y1) and
the coordinates of M as (x2, y2)
in the slope formula.
An equation for AM is
1
y = x + b
4
1
-1 = (- 3) + b
4
3
= b
-1 +
4
1
- = b
4
The equation of the median is y =
EXAMPLE 4
I used the midpoint formula to
calculate the coordinates of M.
I substituted the slope of AM for
m in y = mx + b.
Then I determined the value of b by
substituting the coordinates of A
into the equation and solving for b.
1
1
x - .
4
4
Solving a problem using midpoints
A waste management company is planning to build a landfill in a rural
area. To balance the impact on the two closest towns, the company wants
the landfill to be the same distance from each town. On a coordinate map
of the area, the towns are at A(1, 8) and B(5, 2). Describe all the possible
locations for the landfill.
Wendy’s Solution
x 1 + x 2 y1 + y2
,
b
2
2
1 + 5 8 + 2
,
b
M AB = a
2
2
= (3, 5)
M = a
76
2.1 Midpoint of a Line Segment
I used the midpoint
formula to determine
the coordinates of the
midpoint of AB.
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2.1
y
8
A(1, 8)
6
M(3, 5)
4
2
B(5, 2)
x
-2
0
-2
2
4
6
I drew AB on a grid. I knew
that the points equally far
from A and B lie on the
perpendicular bisector
of AB, so I added this to
my sketch.
Communication
Tip
A perpendicular bisector is
also called a right bisector.
8
(x 1, y1) = A(1, 8)
(x 2, y2) = B(5, 2)
y2 - y1
Slope of AB =
x2 - x1
2 - 8
=
5 - 1
-6
=
4
3
= 2
I needed the slope of the
perpendicular bisector
so that I could write an
equation for it. I used the
slope formula to determine
the slope of AB.
2
The slope of the perpendicular bisector is .
3
Since the perpendicular
bisector is perpendicular
to AB, its slope is the
negative reciprocal of the
slope of AB.
An equation for the perpendicular bisector is
2
y = x + b
3
2
5 = (3) + b
3
5 = 2 + b
5 - 2 = b
3 = b
To determine the value of b,
I substituted the coordinates
of the midpoint of AB into
the equation and solved
for b. This worked because
the midpoint is on the
perpendicular bisector,
even though points A and
B aren’t.
2
x + 3 is the equation of
3
the perpendicular bisector. Possible locations
for the landfill are determined by points that
2
lie on the line with equation y = x + 3.
3
Therefore, y =
NEL
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In Summary
Key Idea
• The coordinates of the midpoint of a line segment are the means
of the coordinates of the endpoints.
Need to Know
y
x1 + x2 y1 + y2
,
b
2
2
can be used to calculate the coordinates
of a midpoint.
• The formula (x, y) = a
• The coordinates of a midpoint can be
used to determine an equation for a
median in a triangle or the perpendicular
bisector of a line segment.
B (x2, y2)
M x1 x2 , y1 y2
2
2
A (x1, y1)
x
冢
0
冢
CHECK Your Understanding
1. Determine the coordinates of the midpoint of each line segment, using
one endpoint and the rise and run. Verify the midpoint by measuring
with a ruler.
y
a)
b)
y
10
10
B(5, 9)
8
8
C(–3, 6)
6
6
4
4
2
2
A(1, 1)
0
a)
4
6
4
2
2
4
-4
78
2
4
2
x
0
-2
0
-2
2
E(–1, 1)
-2
-4
8
2. Determine the coordinates of the midpoint of each line segment.
b)
c)
y
y
6
6
F(5, 5)
G(–2, 6)
4
4
y
6
2
D(2, 1)
x
x
2.1 Midpoint of a Line Segment
6
x
x
-2
0
-2
-4
2
4
6
H(3, –4)
-2
0
-2
2
J(1, –2)
4
6
8
K(7, –2)
-4
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2.1
3. On the design plan for a landscaping project, a straight path runs from
(11, 29) to (53, 9). A lamp is going to be placed halfway along the path.
a) Draw a diagram that shows the path.
b) Determine the coordinates of the lamp on your diagram.
PRACTISING
4. Determine the coordinates of the midpoint of the line segment
with each pair of endpoints.
a) A(- 1, 3) and B(5, 7)
b) J( - 2, 3) and K(3, 4)
c) X(6, - 2) and Y( -2, -2)
d) P(2, -4) and I(-3, 5)
3
5 1
1
e) U a , - b and V a- , - b
2
2
2 2
f ) G(1.5, -2.5) and H(-1, 4)
5. The endpoints of the diameter of a circle are A(-1, 1) and B(2.5, -3).
Determine the coordinates of the centre of the circle.
6. P(- 3, - 1) is one endpoint of PQ. M(1, 1) is the midpoint of PQ.
Determine the coordinates of endpoint Q. Explain your solution.
7. A triangle has vertices at A(2, -2), B( -4, -4), and C(0, 4).
K
a) Draw the triangle, and determine the coordinates of the midpoints
of its sides.
b) Draw the median from vertex A, and determine its equation.
8. A radius of a circle has endpoints O(- 1, 3) and R(2, 2). Determine
the endpoints of the diameter of this circle. Describe any assumptions
you make.
9. A quadrilateral has vertices at P(1, 3), Q(6, 5), R(8, 0), and S(3, -2).
Determine whether the diagonals have the same midpoint.
10. Mayda is sketching her design for a rectangular garden. By mistake,
C
she has erased the coordinates of one of the corners of the garden. As
a result, she knows only the coordinates of three of the rectangle’s
vertices. Explain how Mayda can use midpoints to determine the
unknown coordinates of the fourth vertex of the rectangle.
11. A triangle has vertices at P(7, 7), Q( -3, - 5), and R(5, -3).
A
a) Determine the coordinates of the midpoints of the three sides
of ^PQR.
b) Calculate the slopes of the midsegments of ^PQR.
c) Calculate the slopes of the three sides of ^PQR.
d) Compare your answers for parts b) and c). What do you notice?
12. Determine the equations of the medians of a triangle with vertices
at K(2, 5), L(4, - 1), and M( -2, -5).
NEL
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13. Determine an equation for the perpendicular bisector of a line
segment with each pair of endpoints.
a) C( -2, 0) and D(4, -4)
c) L(-2, - 4) and M(8, 4)
b) A(4, 6) and B(12, -4)
d) Q(-5, 6) and R(1, -2)
14. A committee is choosing a site for a county fair. The site needs to be
located the same distance from the two main towns in the county. On
a map, these towns have coordinates (3, 10) and (13, 4). Determine an
equation for the line that shows all the possible sites for the fair.
15. A triangle has vertices at D(8, 7), E(-4, 1), and F(8, 1). Determine
T
the coordinates of the point of intersection of the medians.
16. In the diagram, ^ A¿B¿C¿ is a reflection
Health Connection
Vegetables, a source of
vitamins and minerals, lower
blood pressure, reduce the risk
of stroke and heart disease,
and decrease the chance of
certain types of cancer.
of ^ ABC. The coordinates of all
vertices are integers.
a) Determine the equation of the line
of reflection.
b) Determine the equations of the
perpendicular bisectors of AA¿, BB¿ ,
and CC¿ .
c) Compare your answers for parts
a) and b). What do you notice?
y
12
line of
reflection
B 8
B
A
-8 C-4
A
0
-4
-8
x
4
8
12
C
17. A quadrilateral has vertices at W( -7, -4), X(-3, 1), Y(4, 2), and
Z(-2, -7). Two lines are drawn to join the midpoints of the
non-adjacent sides in the quadrilateral. Determine the coordinates
of the point of intersection of these lines.
18. Describe two different strategies you can use to determine the
coordinates of the midpoint of a line segment using its endpoints.
Explain how these strategies are similar and how they are different.
Extending
19. A point is one-third of the way from point A(1, 7) to point B(10, 4).
Determine the coordinates of this point. Explain the strategy you used.
20. A triangle has vertices at S(6, 6), T(- 6, 12), and U(0, -12). SM is the
median from vertex S.
a) Determine the coordinates of the point that is two-thirds of the
way from S to M that lies on SM.
b) Repeat part a) for the other two medians, TN and UR.
c) Show that the three medians intersect at a common point. What
do you notice about this point?
d) Do you think the relationship you noticed is true for all triangles?
Explain.
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2.1 Midpoint of a Line Segment
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Length of a Line Segment
GOAL
Determine the length of a line segment.
YOU WILL NEED
• grid paper
• ruler
INVESTIGATE the Math
Some computers can translate a handwritten entry into text by calculating
the lengths of small line segments within the entry and comparing these
lengths to stored information about the lengths of pieces of letters.
?
How can you use the coordinates of the endpoints
of a line segment to determine its length?
A.
Plot two points, A and B, on a grid so that line segment AB is neither
horizontal nor vertical. Join A and B. Then construct a right triangle
that has AB as its hypotenuse.
B.
Write the coordinates of the vertex of the right angle. How are these
coordinates related to the coordinates of the endpoints of the right angle?
C.
Determine the lengths of the horizontal and vertical sides of the right
triangle. How are these lengths related to the coordinates of A and B?
D.
Calculate the length of AB.
E.
Repeat parts A to D for line segment PQ, with endpoints P(x 1, y1) and
Q(x 2, y2).
Reflecting
F.
Does it matter which point is (x 1, y1), and which is (x 2, y2), when
calculating the length of a line segment? Explain.
G.
Describe how to use each of the four coordinates of points P and Q
to determine the length of PQ.
H.
Why do you think the equation for calculating the length of a line
segment is sometimes called the distance formula?
NEL
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APPLY the Math
EXAMPLE 1
Selecting a strategy to calculate the length of a line segment
Determine the length of the line segment with each pair of endpoints.
a) A(2, 6) and B(5, 6)
b) G(⫺7, 8) and H(⫺7, ⫺5)
c) P(⫺4, 7) and Q(3, 1)
Niranjan’s Solution
a)
y
6
A(2, 6)
I noticed that A and B have the same y-coordinate,
so I knew that AB was horizontal. I made a sketch
to check.
B(5, 6)
4
2
x
0
2
4
6
AB = 5 - 2
=3
The length of AB is 3 units.
b)
y
8
G(–7, 8)
6
4
I calculated the difference in the x-coordinates
to determine the length of AB.
I noticed that G and H have the same x-coordinate,
so I knew that GH was vertical. I made a sketch
to check.
2
x
-8 -6 -4
H(–7, –5)
-2
0
-2
-4
-6
GH = 8 - (- 5)
= 13
The length of GH is 13 units.
c)
d = 2(x 2 - x 1)2 + ( y2 - y1)2
PQ = 2[3 - ( - 4)]2 + (1 - 7)2
= 272 + (- 6)2
= 249 + 36
= 285
#
= 9.2
The length of PQ is approximately 9.2 units.
82
2.2 Length of a Line Segment
I calculated the difference in the y-coordinates
to determine the length of GH.
I noticed that the x- and y-coordinates of the
endpoints are different numbers. I knew the line
segment couldn’t be horizontal or vertical.
So, I used the distance formula.
I chose P(⫺4, 7) to be (x1, y1) and Q(3, 1) to be
(x2, y2). I substituted these values into the distance
formula.
I rounded my answer to the nearest tenth of a unit.
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2.2
EXAMPLE 2
Representing distances on a coordinate grid
Winston takes different routes to drive from his home in Toronto
to Carleton University in Ottawa. Sometimes he takes Highway
401 to Prescott, and then Highway 416 to Ottawa. Other times he
drives directly from Toronto to Ottawa along Highway 7. On a
map of southeastern Ontario, with the origin at Windsor and the
coordinates in kilometres, Toronto is at T(301.5, 200.0), Prescott
is at P(580.0, 401.5), and Ottawa is at O(542.0, 474.0).
Approximately how far does Winston drive using each route?
Marla’s Solution
500
y
I made a sketch on grid paper. I saw that I had to
calculate the lengths of TO, TP, and PO, and then
compare TO with TP ⫹ PO.
O(542.0, 474.0)
400
P(580.0, 401.5)
300
200
T(301.5, 200.0)
100
0
x
200
400
600
800
d = 2(x 2 - x 1)2 + (y2 - y1)2
TO = 2(542.0 - 301.5)2 + (474.0 - 200.0)2
= 2240.52 + 274.02
= 257 840.25 + 75 076.00
= 2132 916.25
#
= 365
I used the distance formula to determine the length
of each line segment.
I rounded my first answer to the nearest kilometre
to determine the distance that Winston drives
using the direct route.
TP = 2(580.0 - 301.5)2 + (401.5 - 200.0)2
= 2278.52 + 201.52
= 277 562.25 + 40 602.25
= 2118 164.5
#
= 343.8
I knew that I was going to add the lengths TP and
PO, so I rounded to the nearest tenth of a kilometre
because I planned to use these distances in another
calculation.
PO = 2(542.0 - 580.0)2 + (474.0 - 401.5)2
= 2(- 38.0)2 + 72.52
= 21444.00 + 5256.25
= 26700.25
#
= 81.9
NEL
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TP + PO = 343.8 + 81.9
#
= 426
I added lengths TP and PO to determine the
distance that Winston drives using the
indirect route. I rounded the distance to the
nearest kilometre.
The route from Toronto directly to Ottawa is
approximately 365 km. The route through
Prescott is approximately 426 km.
These distances are estimates because they don’t
take into account turns in the road. Even though
the route along Highways 401 and 416 is longer,
it might be faster since Winston can travel at
a greater speed on multi-lane highways.
EXAMPLE 3
Reasoning to determine the distance between a point and a line
Calculate the distance between point A(6, 5) and the line y = 2x + 3.
Kerry’s Solution
y
10
8
y ⫽ 2x ⫹ 3
B
6
A(6, 5)
4
2
x
-2
0
-2
2
4
6
8
I graphed the line by plotting the y-intercept at
(0, 3). Then I used the slope to determine a second
point on the line. I drew a straight line between
these points. I also plotted point A.
I reasoned that the distance I needed to calculate
was the shortest distance between point A and the
line. I figured that this distance would be measured
on a line through A, perpendicular to y = 2x + 3.
I called the point where the red line and the blue
line intersect point B.
I had to calculate the length of AB. To do this,
I needed the coordinates of B, which meant
that I needed the equation of the red line.
The slope of y = 2x + 3 is 2.
Since the red line is perpendicular to the blue line,
1
The slope of AB is - .
the slope of AB is the negative reciprocal of 2.
2
1
Therefore, y = - x + b is an equation for the red line.
2
1
I substituted the coordinates of A into the equation
5 = - (6) + b
to determine b.
2
5 = -3 + b
8 = b
1
Therefore, y = - x + 8 is the equation of the red line.
2
84
2.2 Length of a Line Segment
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2.2
y = 2x + 3
1
y = - x + 8
2
1
- x + 8 = 2x + 3
2
1
2 a- b x + 2(8) = 2(2x) + 2(3)
2
- x + 16 = 4x + 6
16 - 6 = 4x + x
10 = 5x
2 = x
y = 2(2) + 3
y = 7
To determine the point of intersection, I had
to solve a system of equations.
I used substitution to replace y in the first equation
with the right side of the second equation. Then I
solved for x.
I let x = 2 in the first equation to determine
the value of y.
The coordinates of B are (2, 7).
d = 2(x 2 - x 1)2 + ( y2 - y1)2
dAB = 2(2 - 6)2 + (7 - 5)2
= 2(- 4)2 + 22
= 220
#
= 4.5
I used the distance formula to calculate the
length of AB, where A(x1, y1) = A(6, 5) and
B(x2, y2) = B(2, 7).
The point (6, 5) is about 4.5 units away from
the line y = 2x + 3.
In Summary
Key Idea
y
B(x2, y2)
• The distance, d, between the endpoints of a line segment, A(x1, y1)
and B(x2, y2), can be calculated using the distance formula:
d
d = 2(x2 - x1)2 + ( y2 - y1)2
rise ⫽ y2 ⫺ y1
A(x1, y1)
run ⫽ x2 ⫺ x1
(x2, y1)
x
0
Need to Know
• The Pythagorean theorem is used to develop the distance formula,
by calculating the straight-line distance between two points.
• The distance between a point and a line is the shortest distance between
them. It is measured on a perpendicular line from the point to the line.
NEL
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CHECK Your Understanding
a)
b)
y
6
B(5, 4)
4
1. Determine the length of each line segment.
c)
d)
y
y
D(3, 5)
6
6
E(–2, 6)
4
4
2
2
A(1, 1)
0
2
4
F(3, 2) x
x
0
2 4
C(3, –1)
-4
-2
2
x
2
x
y
4
0
2
-4
-2
0
-2
G(–3, –3) -4
2
4
6
H(7, –3)
2. For each pair of points:
i) Draw the line segment joining the points.
ii) Determine the length of the line segment.
a) P(⫺4, 4) and Q(3, 1)
b) R(2, ⫺1) and S(10, 2)
c) T(3.5, ⫺3) and U(3.5, 11)
d) X(⫺1, 6) and Y(5, 6)
3. A helicopter travelled from Kapuskasing to North Bay. On a map of
Ontario, with the origin at Windsor and the coordinates in kilometres,
Kapuskasing is at K(⫺70, 770) and North Bay is at N(220, 490).
a) Approximately how far did the helicopter travel?
b) What assumption did you make about the route of the helicopter?
PRACTISING
4. Calculate the distance between each pair of points in the diagram
y
4
P
Q
-4
at the left.
a) P and Q
b) Q and R
R
2
T x
-2
0
-2
U -4
2
4
S
c) U and S
d) P and R
e) P and U
f ) Q and T
5. For each pair of points below:
i) Draw the line segment joining the points.
ii) Calculate the length of the line segment.
a) A(2, 6) and B(5, 2)
b) C(⫺3, 4) and D(3, 2)
c) E(⫺6, 8) and F(⫺6, ⫺9)
d) G(0, ⫺7) and H(1, 3)
e) I(⫺3, ⫺3) and J(5, ⫺4)
f ) K(⫺10, ⫺2) and L(6, ⫺2)
6. a) Which line segment(s) for question 5 are vertical? Which are
horizontal? Explain how you know.
b) How can you calculate the length of a vertical or horizontal line
segment without using the distance formula?
7. A coordinate system is superimposed on a billiard table. Gord has
a yellow ball at A(2, 3). He is going to “bank” it off the side rail at
B(6, 5), into the pocket at C(2, 7). How far will the yellow ball travel?
8. Which of these points is closest to point A(⫺3.2, 5.6): B(1.8, ⫺4.3),
K C(0.7, 8.9), or D(⫺7.6, 3.9)? Justify your decision.
86
2.2 Length of a Line Segment
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2.2
9. A forest fire is threatening two small towns, Mordon and Bently. On
a map, the fire is located at (10, ⫺11), the fire hall in Mordon is
located at (26, 77), and the fire hall in Bently is located at (12, ⫺88).
Which fire hall is closer to the fire?
10. In a video game, three animated characters are programmed to run
out of a building at F(1, ⫺1) and head in three different directions.
After 2 s, Animal is at A(22, 18), Beast is at B(⫺3, 35), and Creature
is at C(7, ⫺29). Which character ran farthest?
11. How are the formulas for calculating the length of a line segment
C
and the midpoint of a line segment, using the coordinates of the
endpoints, the same? How are they different?
12. Calculate the distance between each line and the point. Round your
answer to one decimal place.
a) y = 4x - 2, (- 3, 3)
b) y = - x + 5, (- 1, - 2)
c) 2x + 3y = 6, (7, 6)
d) 5x - 2y = 10, (2, 4.5)
13. A new amusement park is going to be built near two major highways.
T
On a coordinate grid of the area, with the scale 1 unit represents 1 km,
the park is located at P(3, 4). Highway 2 is represented by the
equation y = 2x + 5, and Highway 10 is represented by the equation
y = - 0.5x + 2. Determine the coordinates of the exits that must be
built on each highway to result in the shortest road to the park.
14. A coordinate grid is superimposed on the plan of a new housing
development. A fibre-optic cable is being laid to link points A(⫺18, 12),
B(⫺8, 1), C(3, 4), and D(15, 7) in a run beginning at A and ending at
D. If one unit on the grid represents 2.5 m, how much cable is required?
15. A leash-free area for dogs is going to be created in a field behind a
A
recreation centre. The area will be in the shape of an irregular pentagon,
with vertices at (2, 0), (1, 6), (8, 9), (10, 7), and (6, 0). If one unit on
the plan represents 10 m, what length of fencing will be required?
16. Suppose that you know the coordinates of three points. Explain how
you would determine which of the first two points is closer to the third
point. Describe the procedures, facts, and formulas you would use,
and give an example.
Extending
17. ^ ABC has vertices at A(1, 2), B(4, 8), and C(8, 4).
a) ^ ABC is translated so that vertex A¿ is on the x-axis and vertex B¿
is on the y-axis. Determine the coordinates of the translated
triangle, ^ A¿B¿C¿ .
b) ^ DEF has vertices at D(⫺1, 1), E(⫺2, 6), and F(⫺8, 3). Is
^ DEF congruent to ^ ABC ? Justify your answer.
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YOU WILL NEED
• graphing calculator
• grid paper, ruler, and
compass, or dynamic
geometry software
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Equation of a Circle
GOAL
Develop and use an equation for a circle.
INVESTIGATE the Math
When an earthquake occurs, seismographs can be used to record the shock
waves. The shock waves are then used to locate the epicentre of the
earthquake—the point on Earth located directly above the rock movement.
The time lag between the shock waves is used to calculate the distance
between the epicentre and each recording station, which avoids considering
direction.
Career Connection
A geologist studies the physical
structure and processes of
Earth. Professional geologists
work for a wide range of
government agencies, private
firms, non-profit organizations,
and academic instututions.
Tech
Support
For help constructing a circle
and plotting points using
dynamic geometry software,
see Appendix B-34 and B-18.
88
2.3 Equation of a Circle
A seismograph in Collingwood, Ontario, recorded vibrations indicating
that the epicentre of an earthquake was 30 km away.
?
What equation describes the possible locations of the epicentre
of the earthquake, if (0, 0) represents the location of the
seismograph?
A.
Tell why the equation of a circle that has its centre at the origin and
a radius of 30 describes all the possible locations of the epicentre.
B.
Sketch this circle on a grid. Then identify the coordinates of all
its intercepts.
C.
Show that (24, 18), (24, 18), (24,18), and (24, 18) are
possible locations of the epicentre, using
i) your graph
ii) the distance formula
D.
Let point A(x, y) be any possible location of the epicentre. What is the
length of OA? Explain.
E.
Use the distance formula to write an expression for the length of OA.
F.
Use your results for parts D and E to write an equation for the circle
that is centred at the origin. Write your equation in a form that does
not contain a square root. Explain why your equation describes all the
possible locations of the epicentre.
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2.3
Reflecting
G.
If (x, y) is on the circle that is centred at the origin, so are (x, y),
(x, y), and (x, y). How does your equation show this?
H.
How is the equation of a circle different from the equation of a linear
relationship?
I.
What is the equation of any circle that has its centre at the origin and
a radius of r units?
APPLY the Math
EXAMPLE 1
Selecting a strategy to determine the equation of a circle
A stone is dropped into a pond, creating a circular ripple. The radius of the
ripple increases by 4 cm/s. Determine an equation that models the circular
ripple, 10 s after the stone is dropped.
Aurora’s Solution
I named the point where the stone entered the water
(0, 0). I knew that the equation of a circle with centre
(0, 0) and radius r is x 2 + y 2 = r 2.
x2 + y2 = r2
r = (4 cm/s)(10 s)
r = 40 cm
I wanted to determine the radius of the circle at 10 s, so
I multiplied the rate at which the radius increases by 10.
x 2 + y 2 = 402
x 2 + y 2 = 1600
I substituted the value of the radius for r into the equation.
The equation of the circular
ripple is x 2 + y 2 = 1600.
EXAMPLE 2
Selecting a strategy to graph a circle, given the equation of the circle
A circle is defined by the equation x 2 + y 2 = 25. Sketch a graph of this circle.
Francesco’s Solution
x 2 + y 2 = 25
To determine the x-intercepts, let y = 0.
x 2 + 02 = 25
x 2 = 25
2x 2 = ; 125
x = ;5
I decided to determine some points on the circle. I knew
that I could determine the intercepts by setting each
variable equal to 0 and solving for the other variable.
I remembered that there are two possible square roots:
one positive and one negative.
The x-intercepts are located at (5, 0) and ( -5, 0).
NEL
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To determine the y-intercepts, let x = 0.
02 + y 2 = 25
y 2 = 25
2y 2 = ; 125
y = ;5
The y-intercepts are located at (0, 5) and (0, -5).
x 1 + x 2 y1 + y2
,
b
2
2
5 + ( - 5) 0 + 0
M AB = a
,
b
2
2
= (0, 0)
The centre is at (0, 0).
M = a
I solved for y.
Since a circle has symmetry, I reasoned that the
x-intercepts are endpoints of a diameter. Since all the
points on a circle are the same distance from the centre,
the midpoint of this diameter must be the centre.
The line segments that join (0, 0) to each intercept are
radii. Since these are horizontal and vertical lines whose
lengths are 5 units, this circle has a radius of 5 units.
The radius equals 5 units.
y
6 (0, 5)
(–3, 4)
(3, 4)
4
2
(–5, 0)
-2
-6 -4
(–3, –4)
(5, 0) x
0
-2
2
-4
4
6
I plotted the intercepts and then joined them with a
smooth circle. I noticed that (3, 4) is on the circle, and
that points with similar coordinates are too. This makes
sense because a circle with centre (0, 0) is symmetrical
about any line that passes through the origin.
(3, –4)
-6 (0, –5)
EXAMPLE 3
Reasoning to determine the equation of a circle
A circle has its centre at (0, 0) and passes through point (8, 6).
a) Determine the equation of the circle.
b) Determine the other endpoint of the diameter through (8, 6).
Trevor’s Solution
a)
x2 + y2 = r2
82 + ( - 6)2 = r 2
100 = r 2
The equation of the
circle is x 2 + y 2 = 100.
90
2.3 Equation of a Circle
I started with the equation x 2 + y 2 = r 2, since the
circle is centred at the origin. I knew that point (8, 6)
is on the circle, so I substituted 8 for x and 6 for y
into the equation.
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2.3
b) r = 2100
r = 10
12
Q(–8, 6)
8
6
4
-12 -8
8
x2 y2 100
0
-4
I calculated the radius of the circle. Then I drew a sketch,
making sure that the circle passed through 10 and 10
on both the x- and y-axes since 10 is the radius.
y
8
4
-8
I drew the diameter that starts at P(8, 6), passes
through (0, 0), and ends at point Q.
x
Because a circle is symmetrical and PQ is a diameter,
I reasoned that Q has coordinates (8, 6).
6 12
P(8, –6)
-12
The other endpoint of the diameter that passes
through point (8, 6) has coordinates (8, 6).
In Summary
y
Key Idea
• Using the distance formula, you can show that
the equation of a circle with centre (0, 0) and
radius r is x 2 + y 2 = r 2.
P(x, y)
r
0
y x
x
Need to Know
• Every point on the circumference of a circle is the same distance from
the centre of the circle.
• Once you know one point on a circle with centre (0, 0), you can
determine other points on the circle using symmetry. If (x, y) is on
a circle with centre (0, 0), then so are (x, y), (x, y), and (x, y).
CHECK Your Understanding
1. The graph at the right shows a circle with its centre at (0, 0).
a) State the x-intercepts of the circle.
b) State the y-intercepts.
c) State the radius.
d) Write the equation of the circle.
4
x
-8 -4
2. Write the equation of a circle with centre (0, 0) and radius r.
a) r = 3
NEL
b) r = 50
1
c) r = 2
3
y
8
d) r = 400
0
-4
4
8
-8
e) r = 0.25
Chapter 2
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3. For each equation of a circle:
i) Determine the radius.
ii) State the x- and y-intercepts.
iii) Sketch its graph.
a) x 2 + y 2 = 36
b) x 2 + y 2 = 49
c) x 2 + y 2 = 0.04
d) x 2 + y 2 = 169
PRACTISING
4. Use the given information to write an equation for a circle
with centre (0, 0).
a) radius of 11 units
c) y-intercepts (0, 4) and (0, 4)
b) x-intercepts (9, 0) and (9, 0) d) diameter of 12 units
5. A circle has its centre at (0, 0) and passes through the point (8, 15).
a) Calculate the radius of the circle.
b) Write an equation for the circle.
c) Sketch the graph.
6. Which of the following points are on the circle with equation
x 2 + y 2 = 65? Explain.
a) (4, 7)
b) (5, 6)
c) (8, 1)
d) (3, 6)
7. a) Determine the radius of a circle that is centred at (0, 0) and passes
K
through
i) (3, 4)
ii) (5, 0)
iii) (0, 3)
iv) (8, 15)
b) Write the equation of each circle in part a).
c) State the coordinates of two other points on each circle.
d) Sketch the graph of each circle.
8. Write an equation for a circle that models each situation. Assume that
(0, 0) is the centre of the circle in each situation.
a) the possible locations of the epicentre of an earthquake, which is
recorded to be a distance of 144 km from a seismograph station
in Toronto
b) the path of a satellite in a circular orbit at a distance of 19 000 km
from the centre of Earth
c) the rim of a bicycle wheel with a diameter of 69 cm
d) the cross-section of a storm-water tunnel that has a diameter of 2.4 m
9. Don has designed a circular vegetable garden with a diameter of
24 m
92
2.3 Equation of a Circle
24.0 m, as shown in the sketch at the left. He has included a circular
flowerbed, 6.0 m in diameter, at the centre of the garden, as well as
paths that are 1.5 m wide. Don needs to make a plan on grid paper for
the landscape gardeners, who will create the garden. Determine the
equations of all the circles he needs to draw. Assume that the centre of
the garden is (0, 0) and all noncircular gardens are of equal width.
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2.3
10. Two satellites are orbiting Earth. The path of one satellite has the
A
equation x 2 + y 2 = 56 250 000. The orbit of the other satellite is
200 km farther from the centre of Earth. In one orbit, how much
farther does the second satellite travel than the first satellite?
11. A circle has its centre at (0, 0) and passes through point P(5, 12).
a) Determine the equation of the circle.
b) Determine the coordinates of the other endpoint of the diameter
that passes through point P.
12. Determine the equation of a circle that has a diameter with endpoints
(8, 15) and (8, 15).
13. A rock is dropped into a pond, creating a circular ripple. The radius of
the ripple increases steadily at 6 cm/s. A toy boat is floating on the
pond, 2.00 m east and 1.00 m north of the spot where the rock is
dropped. How long does the ripple take to reach the boat?
Career Connection
Engineers design satellites for
communication including
television, the Internet, and
phone systems. Other uses of
satellites include observation
in the area of espionage,
geology, and navigation
such as GPS systems.
14. Points (a, 5) and (9, b) are on the circle x 2 + y 2 = 125. Determine
the possible values of a and b. Round to one decimal place, if
necessary.
15. A satellite orbits Earth on a path with x 2 + y 2 = 45 000 000.
C
Another satellite, in the same plane, is currently located at
(12 504, 16 050). Explain how you would determine whether the
second satellite is inside or outside the orbit of the first satellite.
16. Chanelle is creating a design for vinyl flooring.
T
She uses circles and squares to create the design,
as shown. If the equation of the small circle is
x 2 + y 2 = 16, what are the dimensions of the
large square?
17. What reasons would you use to convince someone that it makes sense
for the equation of a circle with centre (0, 0) and radius r to be
x 2 + y 2 = r 2? Use as many reasons as you can.
Extending
18. Describe the circle with each equation.
a) 9x 2 + 9y 2 = 16
b) (x - 2)2 + ( y + 4)2 = 9
19. A truck with a wide load, proceeding slowly along a secondary road, is
approaching a tunnel that is shaped like a semicircle. The maximum
height of the tunnel is 5.25 m. If the load is 8 m wide and 3.5 m high,
will it fit through the tunnel? Show your calculations, and explain your
reasoning.
NEL
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Study
Q:
Aid
• See Lesson 2.1, Example 1.
• Try Mid-Chapter Review
A:
Questions 1 and 2.
Study
Aid
x 1 + x 2 y1 + y2
,
b . This
2
2
formula shows that the coordinates of the midpoint are the means
of the coordinates of the endpoints.
You can use the midpoint formula M = a
Q:
How do you determine the length of a line segment if you
know the coordinates of the endpoints?
A:
If the endpoints have the same x-coordinate, then the line segment
is vertical. The length of the line segment is the difference in the
y-coordinates of the endpoints. Similarly, if the endpoints have the
same y-coordinate, then the line segment is horizontal. The length of
the line segment is the difference in the x-coordinates of the endpoints.
• See Lesson 2.2,
Examples 1 to 3.
How do you determine the coordinates of the midpoint of a
line segment if you know the coordinates of the endpoints?
• Try Mid-Chapter Review
Questions 6 to 9.
For all types of line segments, including those which are neither
vertical nor horizontal, you can use the distance formula to calculate
its length.
d = 2(x 2 - x 1)2 + ( y2 - y1)2
Study
Aid
Q:
How do you determine the equation of a circle that has
its centre at the origin?
A1:
The equation of a circle with centre (0, 0) is x 2 + y 2 = r 2, where r is
the radius. For example, the equation of a circle with centre (0, 0) and
a radius of 4 units is x 2 + y 2 = 42, or x 2 + y 2 = 16.
A2:
If you only know the coordinates of a point on the circle, you can
substitute these values for x and y and then solve for r. For example,
suppose that you want to determine the equation of a circle that has its
centre at the origin and passes through point (2, 9). You substitute
2 for x and 9 for y.
• See Lesson 2.3,
Examples 1 and 3.
• Try Mid-Chapter Review
Question 11.
x2 y2 85
8
y
4
x
-8 -4
0
-4
4
8
x2 y2 16
-8
22 + ( -9)2 = r 2
4 + 81 = r 2
85 = r 2
The circle has equation x 2 + y 2 = 85.
94
Mid-Chapter Review
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Mid-Chapter Review
PRACTICE Questions
Lesson 2.1
1. Determine the coordinates of the midpoint of
the line segment with each pair of endpoints.
a) (1, 2) and (7, 10)
b) (5, 1) and (2, 9)
c) (0, 4) and (0, 12)
d) (6, 4) and (0, 0)
2. A diameter of a circle has endpoints A(9, 4)
and B(3, 2). Determine the centre
of the circle.
3. Describe all the points that are the same
distance from points A(3, 1) and B(5, 3).
4. A hockey arena is going to be built to serve two
rural towns. On a plan of the area, the towns are
located at (1, 7) and (8, 5). If the arena needs to
be the same distance from both towns,
determine an equation to describe the possible
locations for the arena.
5. ^PQR has vertices at P(12, 4), Q(6, 2), and
R(4, 2).
a) Determine the coordinates of the midpoints
of its sides.
b) Determine the equation of the median from
vertex Q.
c) What is the equation of the perpendicular
bisector of side PQ?
Lesson 2.2
6. Calculate the distance between each pair
of points.
a) (2, 2) and (7, 4)
b) (3, 0) and (8, 5)
c) (2, 9) and (5, 9)
d) (9, 3) and (12, 4)
7. A power line is going to be laid from A(22, 15)
to B(7, 33) to C(10, 18) to D(1, 4). If the units
are metres, what length will the power line be?
8. Determine the distance between point (4, 4)
and the line y = 3x - 4.
NEL
9. Show that ^ ABC has three unequal sides.
A 6 y
4
2
B
x
-4
0
C 4
Lesson 2.3
10. i)
State the coordinates of the centre of the
circle described by each equation below.
ii) State the radius and the x- and y-intercepts
of the circle.
iii) Sketch a graph of the circle.
a) x 2 + y 2 = 169
b) x 2 + y 2 = 2.89
c) x 2 + y 2 = 98
11. Determine the equation of a circle that has its
centre at (0, 0) and passes through each point.
a) (5, 0)
c) (3, 8)
b) (0, 7)
d) (4, 9)
12. A raindrop falls into a puddle, creating a circular
ripple. The radius of the ripple grows at a steady
rate of 5 cm/s. If the origin is used as the
location where the raindrop hits the puddle,
determine the equation that models the ripple
exactly 6 s after the raindrop hits the puddle.
13. Determine whether each point is on, inside, or
outside the circle x 2 + y 2 = 45. Explain your
reasoning.
a) (6, 3)
c) (3, 5)
b) (1, 7)
d) (7, 2)
14. A line segment has endpoints A(6, 7) and B(2, 9).
a) Verify that the endpoints of AB are on the
circle with equation x 2 + y 2 = 85.
b) Determine the equation of the
perpendicular bisector of AB.
c) Explain how you can tell, from its equation,
that the perpendicular bisector goes through
the centre of the circle.
Chapter 2
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Classifying Figures on
a Coordinate Grid
YOU WILL NEED
GOAL
• grid paper and ruler, or
Use properties of line segments to classify two-dimensional
figures.
dynamic geometry software
LEARN ABOUT the Math
A surveyor has marked the corners of a lot where a building is going
to be constructed. The corners have coordinates P(-5,-5), Q( -30, 10),
R(- 5, 25), and S(20, 10). Each unit represents 1 m. The builder wants
to know the perimeter and shape of this building lot.
?
EXAMPLE 1
How can the builder use the coordinates of the corners
to determine the shape and perimeter of the lot?
Connecting slopes and lengths of line segments to classifying a figure
Use analytic geometry to identify the shape of quadrilateral PQRS and
its perimeter.
Anita’s Solution
R
20
Q
S
10
x
-30
-20
0
-10
10
P
-10
96
geometry that uses the xy-axes,
algebra, and equations to
describe relations and positions
of geometric figures
I plotted the points on grid
paper and then joined
them to draw the figure.
y
30
analytic geometry
2.4 Classifying Figures on a Coordinate Grid
20
I saw that the shape of the
building lot looked like a
parallelogram or a rhombus,
but I couldn’t be sure.
I knew that if PQRS was
either of these figures, the
opposite sides would be
parallel. I also knew that if
PQRS was a rhombus, all
the sides would be the
same length.
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2.4
y2 - y1
x2 - x1
Length = 2(x 2 - x 1)2 + ( y2 - y1)2
Slope =
PQ = 2[ - 30 - ( - 5)]2 + [10 - (-5)]2
10 - ( -5)
-30 - (-5)
15
=
-25
3
= 5
25 - 10
m QR =
-5 - (-30)
15
=
25
3
=
5
10 - 25
m RS =
20 - (- 5)
-15
=
25
3
= 5
-5 - 10
m SP =
-5 - 20
-15
=
-25
3
=
5
= 2625 + 225
= 2850
#
= 29.15
The length of PQ is about 29.15 units.
QR = 2[- 5 - ( -30)]2 + (25 - 10)2
= 2625 + 225
= 2850
#
= 29.15
The length of QR is about 29.15 units.
RS = 2[20 - (- 5)]2 + (10 - 25)2
= 2625 + 225
= 2850
#
= 29.15
The length of RS is about 29.15 units.
SP = 2( - 5 - 20)2 + (- 5 - 10)2
= 2625 + 225
= 2850
#
= 29.15
The length of SP is about 29.15 units.
m PQ =
PQ || RS and QR || SP
PQ = QR = RS = SP
Since the opposite sides are parallel and all the side lengths are equal,
PQRS is a rhombus.
#
Perimeter = 4(29.15)
= 116.6
I decided to calculate the
slope and the length of
each side of PQRS.
The slopes of PQ and RS
are the same, so they are
parallel. The slopes of QR
and SP are also the same,
so they are parallel too.
My length calculations
showed that all four sides
are equal.
Communication
Tip
The symbol || is used to replace
the phrase “is parallel to.”
I multiplied the side length
by 4 to calculate the
perimeter.
The building lot is a rhombus. Its perimeter measures about 116.6 m.
Reflecting
A.
Why could Anita not rely completely on her diagram to determine
the shape of the quadrilateral?
B.
Why did Anita need to calculate the slopes and lengths of all the sides
to determine the shape of the building lot? Explain.
NEL
Chapter 2
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APPLY the Math
EXAMPLE 2
Reasoning about lengths and slopes to classify a triangle
A triangle has vertices at A( -1, - 1), B(2, 0), and C(1, 3). What type
of triangle is it? Justify your decision.
Angelica’s Solution
y
C(1, 3)
4
2
B(2, 0) x
A(–1, –1)
-2
0
2
4
I drew a diagram of the triangle on grid paper. I
thought that the triangle might be isosceles since
AB and BC look like they are the same length.
The triangle might also be a right triangle since
∠ABC looks like it might be 90°.
AB = 2[2 - (- 1)]2 + [0 - (- 1)]2
= 232 + 12
= 210
#
= 3.2
BC = 2(1 - 2)2 + (3 - 0)2
= 2( - 1)2 + 32
= 210
#
= 3.2
AB and BC are the same length, so ^ ABC is isosceles.
y2 - y1
Slope =
x2 - x1
0 - (- 1)
m AB =
2 - (- 1)
1
=
3
3 - 0
m BC =
1 - 2
3
=
-1
= -3
The slopes of AB and BC are negative reciprocals, so
AB ⬜ BC. This means that ^ ABC is a right triangle.
^ ABC is an isosceles right triangle, with AB = BC
and ∠ ABC = 90°.
98
2.4 Classifying Figures on a Coordinate Grid
To determine the type of triangle, I had to know
the lengths of the sides. To check my isosceles
prediction, I used the distance formula to determine
the lengths of AB and BC. I rounded each answer
to one decimal place.
To determine if the triangle has a right angle,
I had to determine if a pair of sides are perpendicular.
I did this by comparing their slopes. I calculated the
slopes of the sides that looked perpendicular,
AB and BC.
Communication
Tip
The symbol ⬜ is used to
replace the phrase “is
perpendicular to.”
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2.4
EXAMPLE 3
Solving a problem using properties of line segments
Tony is constructing a patterned concrete patio that is in the shape of an
isosceles triangle, as requested by his client. On his plan, the vertices of the
triangle are at P(2, 1), Q(5, 7), and R(8, 4). Each unit represents 1 m.
a) Confirm that the plan shows an isosceles triangle.
b) Calculate the area of the patio.
Tony’s Solution
a)
y
8
Q(5, 7)
6
4
R(8, 4)
I made a sketch of the triangle. It looks isosceles
since PQ and RP appear to be the same length.
2
x
P(2, 1)
0
2
4
6
8
Length = 2(x 2 - x 1)2 + ( y2 - y1)2
PQ = 2(5 - 2)2 + (7 - 1)2
= 245
#
= 6.7
I used the distance formula to calculate the lengths
of the sides of the triangle. I saw that PQ is the
same length as RP, so the triangle is isosceles.
RP = 2(2 - 8)2 + (1 - 4)2
= 245
#
= 6.7
QR = 2(8 - 5)2 + (4 - 7)2
= 218
#
= 4.2
Since PQ = RP, the triangle is isosceles.
b)
y
8
Q(5, 7)
M
6
4
R(8, 4)
2
x
P(2, 1)
0
NEL
2
4
6
8
I knew that I needed the lengths of the base and
height to calculate the area of the triangle, since
base * height
area =
.
2
I remembered that, in an isosceles triangle, the
median from the vertex where the two equal sides
meet is perpendicular to the side that is opposite
this vertex. PM is the height of the triangle and QR
is its base.
Chapter 2
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Page 100
Midpoint of QR is
x 1 + x 2 y1 + y2
,
b
M = a
2
2
5 + 8 7 + 4
,
b
= a
2
2
= (6.5, 5.5)
PM = 2(6.5 - 2)2 + (5.5 - 1)2
= 220.25 + 20.25
= 240.5
#
= 6.4
Area of ^ PQR =
I calculated M, the midpoint of QR, so I
could use it to determine the length of PM.
I used the distance formula to calculate the
length of PM. I already knew that the length
of QR is 218. PM is the height of the triangle,
and QR is the base.
QR * PM
2
218 * 240.5
2
= 13.5
=
I calculated the area of the triangle using the
exact values to minimize the rounding error.
The triangular patio has an area of 13.5 m2.
I checked my calculations by plotting the vertices
of the triangle using dynamic geometry software.
Then I constructed the triangle and its interior.
I measured the lengths of the sides and determined
the area. The scale in this sketch is 1 unit = 1 cm
instead of 1 unit = 1 m. Since the numbers were
the same, however, I knew that my calculations
were correct.
Tech
Support
Do not change the scale of the
dynamic geometry software
grid, because this will make
the unit length different from
1 cm.
100
2.4 Classifying Figures on a Coordinate Grid
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2.4
In Summary
Key Idea
• When a geometric figure is drawn on a coordinate grid, the coordinates
of its vertices can be used to calculate the slopes and lengths of the line
segments, as well as the coordinates of the midpoints.
Need to Know
• Triangles and quadrilaterals can be classified by the relationships
between their sides and their interior angles.
Triangles
x
o
o o
equilateral
triangle
o
o
isosceles
triangle
scalene
triangle
x
isosceles right
triangle
right
triangle
Quadrilaterals
x
x
o
o
x
parallelogram
o
o
x
rectangle
rhombus
square
x
o
o
irregular
quadrilateral
trapezoid
x
o
o
isosceles
trapezoid
kite
• To solve a problem that involves a geometric figure, it is a good idea
to start by drawing a diagram of the situation on a coordinate grid.
• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals.
CHECK Your Understanding
Round all answers to two decimal places, where necessary.
1. Show that the line segment joining points P(1, 4) and Q(5, 5) is
parallel to the line segment joining points R(3, - 4) and S(7, - 3).
2. Show that TU, T( - 1, 7) and U(3, 5), is perpendicular
to VW, V( - 4, 1) and W( - 1, 7).
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Chapter 2
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3. The sides of quadrilateral ABCD have the following slopes.
Side
AB
BC
CD
AD
Slope
-5
-
1
7
-5
-
1
7
What types of quadrilateral could ABCD be? What other information
is needed to determine the exact type of quadrilateral?
4. ^DEF has vertices at D(- 3, -4), E(-2, 4), and F(5, -5).
a) Show that ^DEF is isosceles.
b) Determine the length of the median from vertex D.
c) Show that this median is perpendicular to EF.
PRACTISING
5. The lengths of the sides in a quadrilateral are PQ = 4.5 units,
QR = 4.5 units, RS = 4.5 units, and SP = 4.5 units. What types
of quadrilateral could PQRS be? What other information is needed
to determine the exact type of quadrilateral?
6. The following points are the vertices of triangles. Predict whether each
triangle is scalene, isosceles, or equilateral. Then draw the triangle on a
coordinate grid and calculate each side length to check your prediction.
a) A(3, 3), B(-1, 2), C(0, -2) c) D(2, -3), E( - 2, -4), F(6, - 6)
b) G(-1, 3), H(-2, -2), I(2, 0) d) J(2, 5), K(5, -2), L(- 1, - 2)
7. P(-7, 1), Q(-8, 4), and R(-1, 3) are the vertices of a triangle. Show
that ^ PQR is a right triangle.
8. A triangle has vertices at L(-7, 0), M(2, 1), and N(-3, 5). Verify that
it is a right isosceles triangle.
9. a) How can you use the distance formula to decide whether points
P(-2, -3), Q(4, 1), and R(2, 4) form a right triangle? Justify
your answer.
b) Without drawing any diagrams, explain which sets of points
are the vertices of right triangles.
i) S(-2, 2), T(-1, –2), U(7, 0)
ii) X(3, 2), Y(1, -2), Z(-3, 6)
iii) A(5, 5), B(3, 8), C(8, 7)
10. A quadrilateral has vertices at W( -3, 2), X(2, 4), Y(6, -1), and
Z(1, -3).
a) Determine the length and slope of each side of the quadrilateral.
b) Based on your calculations for part a), what type of quadrilateral
is WXYZ? Explain.
c) Determine the difference in the lengths of the two diagonals
of WXYZ.
102
2.4 Classifying Figures on a Coordinate Grid
NEL
2.4
11. A polygon is defined by points R(- 5, 1), S(5, 3), T(2, -1), and
U(- 8, - 3). Show that the polygon is a parallelogram.
12. A quadrilateral has vertices at A(- 2, 3), B(-2, -2), C(2, 1), and
D(2, 6). Show that the quadrilateral is a rhombus.
13. a) Show that EFGH, with vertices at E( -2, 3), F(2, 1), G(0, - 3),
K
and H( - 4, - 1), is a square.
b) Show that the diagonals of EFGH are perpendicular to each other.
14. The vertices of quadrilateral PQRS are at P(0, -5), Q( - 9, 2), R(- 5, 8),
and S(4, 2). Show that PQRS is not a rectangle.
15. A square is a special type of rectangle. A square is also a special type
C
of rhombus. How would you apply these descriptions of a square
when using the coordinates of the vertices of a quadrilateral to determine
the type of quadrilateral? Include examples in your explanation.
16. Determine the type of quadrilateral described by each set of vertices.
Give reasons for your answers.
a) J(- 5, 2), K(- 1, 3), L(-2, - 1), M(-6, -2)
b) E( - 5, - 4), F( - 5, 1), G(7, 4), H(7, -1)
c) D(- 1, 3), E(6, 4), F(4, -1), G( -3, - 2)
d) P(- 5, 1), Q(3, 3), R(4, - 1), S( - 4, - 3)
17. A surveyor is marking the corners of a building lot. If the corners have
A
coordinates A(- 5, 4), B(4, 9), C(9, 0), and D(0, - 5), what shape is
the building lot? Include your calculations in your answer.
18. Points P(4, 12), Q(9, 14), and R(13, 4) are three vertices of a rectangle.
T
a) Determine the coordinates of the fourth vertex, S.
b) Briefly describe how you found the coordinates of S.
c) Predict whether the lengths of the diagonals of rectangle PQRS are
the same length. Check your prediction.
19. Suppose that you know the coordinates of the vertices of a
quadrilateral. What calculations would help you determine if the
quadrilateral is a special type, such as a parallelogram, rectangle,
rhombus, or square? How would you use the coordinates of the
vertices in your calculations? Organize your thoughts in a flow chart.
Extending
20. a) Show that the midpoints of any pair of sides in a triangle are two
of the vertices of another triangle, which has dimensions that are
exactly one-half the dimensions of the original triangle and a side
that is parallel to a side in the original triangle.
b) Show that the midpoints of the sides in any quadrilateral are
the vertices of a parallelogram.
NEL
Chapter 2
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2.5
GOAL
Use analytic geometry to verify properties of geometric figures.
dynamic geometry software
-6 M
-2 0
-2
-4
-6
-8
-10
-12
y
J
Page 104
Verifying Properties
of Geometric Figures
YOU WILL NEED
• grid paper and ruler, or
12
P(–7, 9) 10
8
6
4
2
10:40 AM
Q(9, 11)
LEARN ABOUT the Math
K
x
2 4 6 8
Carlos has hired a landscape designer to give him some ideas for
improving his backyard, which is a quadrilateral. The designer’s plan on a
coordinate grid shows a lawn area that is formed by joining the midpoints
of the adjacent sides in the quadrilateral. The four triangular areas will be
gardens.
R(9, –1)
?
How can Carlos verify that the lawn area is a parallelogram?
L
S(1, –11)
EXAMPLE 1
Proving a conjecture about a geometric figure
Show that the midsegments of the quadrilateral, with vertices at
P(7, 9), Q(9, 11), R(9, 1), and S(1, 11), form a parallelogram.
Ed’s Solution: Using slopes
- 7 + 9 9 + 11
,
b = (1, 10).
2
2
9 + 9 11 + ( - 1)
K has coordinates a
,
b = (9, 5).
2
2
9 + 1 - 1 + ( - 11)
L has coordinates a
,
b = (5, -6).
2
2
1 + (- 7) - 11 + 9
M has coordinates a
,
b = (-3, - 1).
2
2
midsegment of a quadrilateral
a line segment that connects
the midpoints of two adjacent
sides in a quadrilateral
J has coordinates a
5 - 10
9 - 1
= - 0.625
I used the midpoint formula to
determine the coordinates of the
midpoints of PQ, QR, RS, and SP,
which are J, K, L, and M.
m JK =
m LM =
- 1 - (-6)
-3 - 5
= - 0.625
I needed to show that JK is parallel to
LM and that KL is parallel to MJ.
-6 - 5
m KL =
5 - 9
= 2.75
10 - ( -1)
m MJ =
1 - -3
= 2.75
I used the slope formula,
y2 - y1
m =
, to calculate the slopes
x2 - x1
of JK, KL, LM, and MJ.
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2.5 Verifying Properties of Geometric Figures
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m JK = m LM and m KL = m MJ
JK || LM and KL || MJ
Quadrilateral JKLM is a parallelogram.
I saw that the slopes of JK and LM are
the same and the slopes of KL and MJ
are the same. This means that the
opposite sides in quadrilateral JKLM are
parallel. So quadrilateral JKLM must be
a parallelogram.
Grace’s Solution: Using properties of the diagonals
- 7 + 9 9 + 11
,
b = (1, 10).
2
2
9 + 9 11 + (- 1)
K has coordinates a
,
b = (9, 5).
2
2
9 + 1 - 1 + (-11)
L has coordinates a
,
b = (5, - 6).
2
2
1 + ( -7) -11 + 9
M has coordinates a
,
b = (-3, -1).
2
2
J has coordinates a
1 + 5 10 + (-6)
,
b = (3, 2).
2
2
9 + (-3) 5 + ( -1)
The midpoint of KM is a
,
b = (3, 2).
2
2
The midpoint of JL is a
12
P(–7, 9) 10
8
6
4
2
0
-6 -4
M(–3, –1) -2
-4
-6
-8
-10
-12
y
J(1, 10)
I calculated the coordinates of points
J, K, L, and M, the midpoints of the
sides in quadrilateral PQRS.
Then I calculated the midpoints of the
diagonals JL and KM.
I discovered that both diagonals have
the same midpoint, so they must
intersect at this point.
Q(9, 11)
K(9, 5)
(3, 2)
2
6 8
x
The diagonals of the quadrilateral
bisect each other since they have the
same midpoint.
This means that JKLM must be
a parallelogram.
R(9, –1)
L(5, –6)
S(1, –11)
JKLM is a parallelogram.
Reflecting
A.
How is Ed’s strategy different from Grace’s strategy?
B.
What is another strategy you could use to show that JKLM
is a parallelogram?
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APPLY the Math
EXAMPLE 2
Selecting a strategy to verify a property of a triangle
A triangle has vertices at P(2, 2), Q(1, 3), and R(4, 1). Show that
the midsegment joining the midpoints of PQ and PR is parallel to QR and
half its length.
Andrea’s Solution: Using slopes and lengths of line segments
y
4
P(–2, 2)
Q(1, 3)
2
x
-4 -2
0
2
-2
6
R(4, –1)
The midpoint of PQ is
-2 + 1 2 + 3
Ma
,
b = M( - 0.5, 2.5).
2
2
The midpoint of PR is
- 2 + 4 2 + ( - 1)
Na
,
b = N(1, 0.5).
2
2
P(–2, 2)
-4
-2
I determined the midpoints of PQ and PR. I used
M for the midpoint of PQ and N for the midpoint
of PR.
y
4
M
2
Q(1, 3)
I drew the line segment that joins M to N.
x
0
-2
m QR =
I drew a diagram of the triangle.
N2
6
R(4, –1)
-1 - 3
4 - 1
4
3
0.5 - 2.5
m MN =
1 - ( - 0.5)
-2
=
1.5
= -
= -
I knew that the slopes of QR and MN would be
the same if QR is parallel to MN.
I calculated the slopes of QR and MN.
-2
2
4
I multiplied
by to get - . The slopes are
1.5
2
3
the same, so MN is parallel to QR.
4
3
MN || QR
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QR = 2(4 - 1)2 + (- 1 - 3)2
= 29 + 16
= 225
= 5
MN = 2[1 - (- 0.5)]2 + (0.5 - 2.5)2
Next, I calculated the lengths of QR and MN. The
length of MN is exactly one-half the length of QR.
= 22.25 + 4
= 26.25
= 2.5
1
MN = QR
2
The midsegment that joins the midpoints of PQ and
PR is parallel to QR and one-half its length.
I verified my calculations using dynamic geometry
software. I chose a scale where 1 unit 1 cm.
I constructed the triangle and the midsegment MN.
Then I measured the lengths and slopes of MN and
QR. My calculations were correct.
EXAMPLE 3
Reasoning about lines and line segments to verify a property of a circle
Show that points A(10, 5) and B(2, 11) lie on the circle with equation
x 2 + y 2 = 125. Also show that the perpendicular bisector of chord AB
passes through the centre of the circle.
Drew’s Solution
r = 2125
#
r = 11.2
The intercepts are located at (0, 11.2), (0, 11.2),
(11.2, 0), and (11.2, 0).
Left Side
Right Side Left Side
x 2 + y2
x 2 + y2
125
= 102 + 52
= 22 + (-11)2
= 125
= 125
I knew that x 2 + y 2 = 125 is the equation of
a circle with centre (0, 0) since it is in the form
x 2 + y 2 = r 2.
I calculated the radius and used this value
to determine the coordinates of the intercepts.
Right Side
125
I substituted the coordinates of points A and B into
the equation of the circle to show that A and B are
on the circle.
Points A(10, 5) and B(2, 11) lie on the circle.
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y 2
x y2 125
12
8
I used the intercepts to sketch the circle. I marked
points A and B on the circle.
A(10, 5)
4
x
-12 -8 -4
0
4
-4
8
12
-8
-12
B(2, –11)
The midpoint of AB is
10 + 2 5 + ( - 11)
Ma
,
b = M(6, - 3).
2
2
I determined the midpoint and marked it on my
sketch. I called the midpoint M. Then I sketched the
perpendicular bisector of AB.
y 2
x y2 125
To write an equation for the perpendicular bisector,
I had to know its slope and the coordinates of a
point on it. I already knew that the midpoint
M(6, 3) is on the perpendicular bisector.
12
8
A(10, 5)
4
x
-12 -8 -4
0
-8
-12
m AB =
8
-4
12
M(6, –3)
B(2, –11)
y2 - y1
x2 - x1
- 11 - 5
=
2 - 10
-16
-8
The slope of chord AB is 2.
=
To determine the slope of the perpendicular
bisector, I had to calculate the slope of AB.
I knew that the slope of the perpendicular bisector
1
is the negative reciprocal of 2, which is - .
2
1
The slope of the perpendicular bisector is - .
2
An equation for the perpendicular bisector is
1
y = - x + b.
2
Since M(6, 3) lies on the perpendicular bisector,
1
- 3 = - (6) + b
2
-3 = -3 + b
0 = b
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2.5 Verifying Properties of Geometric Figures
I wrote the equation in the form y = mx + b. Then
I substituted the coordinates of M into the
equation to determine the value of b.
Since b = 0, the line goes through (0, 0), which is
the centre of the circle.
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The equation of the perpendicular bisector of
1
chord AB is y = - x. The y-intercept is 0.
2
The line passes through (0, 0), which is the centre
of the circle.
I verified my calculations using dynamic geometry
software. I constructed the circle, the chord, and
the perpendicular bisector of the chord. The sketch
confirmed that the perpendicular bisector passes
through the centre of the circle.
In Summary
Key Idea
• When you draw a geometric figure on a coordinate grid, you can verify
many of its properties using the properties of lines and line segments.
Need to Know
• You can use the midpoint formula to determine whether a point bisects
a line segment.
• You can use the formula for the length of a line segment to calculate
the lengths of two or more sides in a geometric figure so that you can
compare them.
• You can use the slope formula to determine whether the sides
in a geometric figure are parallel, perpendicular, or neither.
y
10
CHECK Your Understanding
1. Show that the diagonals of quadrilateral
ABCD at the right are equal in length.
B(–2, 6)
A(–6, 4)
2. Show that the diagonals of quadrilateral
and R(5, 2). Show that the median from
vertex Q is the perpendicular bisector of PR.
NEL
K(9, 8)
8
6
6 J(3, 6)
4
4
2
2
L(11, 2)
x
JKLM at the far right are perpendicular.
3. ^PQR has vertices at P(2, 1), Q(1, 5),
y
8
-8 -6
-4 -2
D(–3, –2)
0
C(1, 0)
0
-2
-2
-4
-4
2
4
6
8
x
10
M(3, –4)
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PRACTISING
4. A rectangle has vertices at J(10, 0), K(8, 6), L(12, 6), and
M(6, 12). Show that the diagonals bisect each other.
5. A rectangle has vertices at A(6, 5), B(12, 1), C(8, 13), and
K D(10, 7). Show that the diagonals are the same length.
6. Make a conjecture about the type of quadrilateral shown in question 1. Use
C
analytic geometry to explain why your conjecture is either true or false.
7. Make a conjecture about the type of quadrilateral shown in question 2. Use
analytic geometry to explain why your conjecture is either true or false.
8. A triangle has vertices at D(5, 4), E(1, 8), and F(1, 2). Show that
the height from D is also the median from D.
9. Show that the midsegments of a quadrilateral with vertices at
P(2, 2), Q(0, 4), R(6, 3), and S(8, 1) form a rhombus.
10. Show that the midsegments of a rhombus with vertices at R(5, 2),
S(1, 3), T(2, 1), and U(6,2) form a rectangle.
11. Show that the diagonals of the rhombus in question 10 are
perpendicular and bisect each other.
12. Show that the midsegments of a square with vertices at A(2, 12),
B(10, 8), C(6, 4), and D(6, 0) form a square.
13. a) Show that points A(4, 3) and B(3, 4) lie on x 2 + y 2 = 25.
b) Show that the perpendicular bisector of chord AB passes through
the centre of the circle.
14. A trapezoid has vertices at A(1, 2), B(2, 1), C(4, 2), and D(2, 0).
A
a) Show that the line segment joining the midpoints of BC and AD
is parallel to both AB and DC.
b) Show that the length of this line segment is half the sum
of the lengths of the parallel sides.
15. ^ ABC has vertices at A(3, 4), B(2, 0), and C(5, 0). Prove that the
T
area of the triangle formed by joining the midpoints of ^ ABC is
one-quarter the area of ^ ABC.
16. Naomi claims that the midpoint of the hypotenuse of a right triangle
is the same distance from each vertex of the triangle. Create a flow
chart that summarizes the steps you would take to verify this property.
Extending
17. Show that the intersection of the line segments joining the midpoints
of the opposite sides of a square is the same point as the midpoints
of the diagonals.
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Exploring Properties
of Geometric Figures
GOAL
Investigate intersections of lines or line segments within triangles
and circles.
YOU WILL NEED
• grid paper and ruler, or
dynamic geometry software
EXPLORE the Math
When Lucy drew the medians in a triangle, she noticed that they
intersected at the same point. She recalled that this is the centroid, or
balance point, which is the centre of mass for the triangle.
?
centroid
What other properties of triangles and circles are determined
by the intersection of lines?
Investigating Triangles
A.
Construct a triangle, and label the vertices A, B, and C. Construct
the perpendicular bisector of each side of ^ABC.
B.
Label the intersection of the perpendicular bisectors O. This is the
circumcentre of ^ ABC. Construct a circle with its centre at O and
radius OA. What do you notice?
C.
Repeat parts A and B for other triangles, including some obtuse
triangles. Is the result always the same? Explain.
D.
Construct a new triangle, and draw the altitude from each vertex. The
intersection of the altitudes is the orthocentre of the triangle.
E.
Repeat part D for other triangles. Is the intersection of the altitudes
always inside the triangle? Explain.
F.
Copy and complete this table to summarize what you know about
intersecting lines in triangles.
Type of Triangle
Centre
Type of Intersecting
Lines
Special Property
circumcentre
the centre of the circle that
passes through all three vertices
of a triangle; the circumcentre is
the same distance from all three
vertices
altitude
a line segment that represents
the height of a polygon, drawn
from a vertex of the polygon
perpendicular to the opposite side
orthocentre
the point where the three
altitudes of a triangle intersect
Diagram
centroid
circumcentre
orthocentre
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Investigating Circles
G.
Construct a circle, and draw two chords, JK and LM, that intersect
inside the circle. Label the intersection point O. Measure the lengths
of JO, OK, LO, and OM. Calculate the products JO ⫻ OK and
LO ⫻ OM. Comment on your results.
H.
Repeat part G for other pairs of chords and other circles. Include some
examples with chords that intersect outside the circle.
I.
Copy and complete this table to summarize what you know about
intersecting lines in circles.
Location of Intersection
Property
Diagram
inside the circle
outside the circle
Reflecting
J.
In what type of triangle do the medians, perpendicular bisectors, and
altitudes coincide?
K.
Does the size of a circle or the location of the intersecting chords affect
the property you observed? Explain.
In Summary
Key Idea
• Some properties of two-dimensional figures are determined
by the intersection of lines.
Properties Determined by the Intersection
of Lines or Line Segments
Figure
triangle
Diagram
centroid
All the medians intersect at the same point,
called the centroid.
All the perpendicular bisectors intersect at the
same point, called the circumcentre. The three
vertices of the triangle are the same distance
from this point.
circumcentre
B
O
A
All the altitudes intersect at the same point,
called the orthocentre.
A
C
B orthocentre
A⬘
C⬘
H
C
B⬘
(continued)
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2.6
circle
When two chords intersect, the products
of their segments are equal.
J L
O
J
K
O
K
M
L
M
JO * OK = LO * OM
Need to Know
• In an equilateral triangle, the medians, perpendicular bisectors, and altitudes
intersect at the same point.
FURTHER Your Understanding
1. ^ ABC has vertices at A(5, 1), B(⫺2, 0), and C(4, 8). Determine the
coordinates of the point that is the same distance from each vertex.
2. Show how you know that a median divides a triangle into two smaller
triangles that have the same area.
3. a) AB and CD have endpoints at A(2, 9), B(7, –6), C(7, 6), and
D(2, –9), and they intersect at E(5, 0). Use the products of the
lengths of line segments to show that AB and CD are chords in
the same circle.
b) A bricklayer is constructing a circular arch with the dimensions
shown. Use intersecting chords to determine the radius of the
circle he will use.
H
0.5 m
A
M
2.0 m
B
4. Some similar figures that are constructed on the sides of a right
triangle have an area relationship like the Pythagorean theorem.
a) Construct a right triangle with perpendicular sides that are 6 cm
and 8 cm and a hypotenuse that is 10 cm.
b) Construct each figure in the table shown on the next page on all
three sides of the right triangle. Calculate the areas of the similar
figures. Copy and complete the table.
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Similar
Figures
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A1 ⴝ Area on
6 cm Side
(cm2)
A2 ⴝ Area on
8 cm Side
(cm2)
A1 ⴙ A2
(cm2)
Area on
Hypotenuse
(cm2)
square
semicircle
rectangle
equilateral triangle
right triangle
parallelogram
c) Change the dimensions of the right triangle to investigate whether
this has any effect on the area relationships for each figure.
d) Explain what you have discovered.
YOU WILL NEED
• grid paper
• ruler
Curious Math
The Nine-Point Circle
A circle that passes through nine different points can
be constructed in every triangle. These points can
always be determined using the same strategy.
1. On a grid, draw a triangle with vertices at
P(1, 11), Q(9, 8), and R(1, 2). Determine the midpoints of the sides,
and mark each midpoint with a blue dot.
2. Determine the equation of each altitude. Then determine the
coordinates of the point where the altitude meets the side that is
opposite each vertex. Mark these points on your diagram in red.
3. Determine the coordinates of the orthocentre (the point where all three
altitudes intersect). Then, for each altitude, determine the coordinates
of the midpoint of the line segment that joins the orthocentre to the
vertex. Mark these midpoints on your diagram in green.
4. Determine the coordinates of the circumcentre (the point where all
three perpendicular bisectors intersect). Determine the midpoint of
the line segment that joins the circumcentre to the orthocentre. Mark
this midpoint with the letter N. This is the centre of your nine-point
circle. Draw the circle.
5. Identify how each of the points that lie on the nine-point circle can
be determined for any triangle.
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A1 ⴝ Area on
6 cm Side
(cm2)
A2 ⴝ Area on
8 cm Side
(cm2)
A1 ⴙ A2
(cm2)
Area on
Hypotenuse
(cm2)
square
semicircle
rectangle
equilateral triangle
right triangle
parallelogram
c) Change the dimensions of the right triangle to investigate whether
this has any effect on the area relationships for each figure.
d) Explain what you have discovered.
YOU WILL NEED
• grid paper
• ruler
Curious Math
The Nine-Point Circle
A circle that passes through nine different points can
be constructed in every triangle. These points can
always be determined using the same strategy.
1. On a grid, draw a triangle with vertices at
P(1, 11), Q(9, 8), and R(1, 2). Determine the midpoints of the sides,
and mark each midpoint with a blue dot.
2. Determine the equation of each altitude. Then determine the
coordinates of the point where the altitude meets the side that is
opposite each vertex. Mark these points on your diagram in red.
3. Determine the coordinates of the orthocentre (the point where all three
altitudes intersect). Then, for each altitude, determine the coordinates
of the midpoint of the line segment that joins the orthocentre to the
vertex. Mark these midpoints on your diagram in green.
4. Determine the coordinates of the circumcentre (the point where all
three perpendicular bisectors intersect). Determine the midpoint of
the line segment that joins the circumcentre to the orthocentre. Mark
this midpoint with the letter N. This is the centre of your nine-point
circle. Draw the circle.
5. Identify how each of the points that lie on the nine-point circle can
be determined for any triangle.
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Using Coordinates
to Solve Problems
YOU WILL NEED
• grid paper
• ruler
GOAL
Use properties of lines and line segments to solve problems.
LEARN ABOUT the Math
Rebecca is designing a parking lot. A tall mast
light will illuminate the three entrances, which
will be located at points A, B, and C. Rebecca
needs to position the lamp so that it illuminates
each entrance equally.
Parking Lot Entrances
24 N
20
A(–8, 14)
16
12
B(–4, 8)
C(18, 10)
8
4
?
W
How can Rebecca determine
the location of the lamp?
EXAMPLE 1
-16 -12 -8 -4
E
0
S 4
8
12 16 20
Solving a problem using a triangle property
Determine the location of the lamp in the parking lot that Rebecca is designing.
Jack’s Solution
The lamp should be placed the same
distance from all three vertices of ^ ABC.
24 N
If the lamp is the same distance from all three vertices,
I reasoned that it would be at the centre of a circle that
passes through all three vertices. I remembered that this
point occurs where the perpendicular bisectors of the
sides of the triangle intersect.
20
A(–8, 14)
16
C(18, 10)
12
B(–4, 8)
8
4
W
-16 -12 -8 -4
E
0
S 4
8
12 16 20
The midpoint of AB is
- 8 + ( - 4) 14 + 8
a
,
b = (- 6, 11).
2
2
NEL
I decided to determine the perpendicular bisectors of
AB and BC. I started with AB.
To write an equation, I needed the slope and one point
on the perpendicular bisector. I knew that the midpoint
of AB would be on the perpendicular bisector, so I
calculated this first.
Chapter 2
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8 - 14
- 4 - (- 8)
-6
=
4
3
= 2
The slope of the perpendicular bisector
2
of AB is .
3
2
An equation is y = x + b.
3
2
11 = (-6) + b
3
11 = - 4 + b
15 = b
The equation of the perpendicular
2
bisector of AB is y = x + 15.
3
m AB =
The midpoint of BC is
- 4 + 18 8 + 10
,
b = (7, 9).
a
2
2
10 - 8
m BC =
18 - ( - 4)
2
=
22
1
=
11
The slope of the perpendicular bisector
of BC is - 11.
Page 116
Because the bisector is perpendicular to AB, its
slope is the negative reciprocal of the slope of AB.
I calculated the slope of AB. Then I figured out
the negative reciprocal to get the slope of the
perpendicular bisector.
2
I wrote an equation of a line with slope , then
3
I substituted the coordinates of the midpoint into
the equation to determine the value of b.
I did the same calculations for BC.
The negative reciprocal of
1
is 11.
11
An equation is y 11x b.
9 11(7) b
9 77 b
86 b
The equation of the perpendicular
bisector of BC is y 11x 86.
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2.7
2
x + 15
3
y = - 11x + 86
At the point of intersection,
2
x + 15 = - 11x + 86
3
35
x = 71
3
35
3a b x = 3(71)
3
35x = 213
213
x =
35
#
x = 6.09
2 213
b + 15
y = a
3 35
142
+ 15
y =
35
#
y = 19.06
To determine where the two perpendicular bisectors
intersect, I set up their equations as a system of
equations. I used the method of substitution to
solve this system of equations.
y =
First, I determined x. Then I substituted the value of x
into the equation of the perpendicular bisector of AB
to determine the value of y. I used the fractional value
for x to minimize any rounding error.
36 N
32
28
24
20
A(–8, 14)
16
(6, 19)
C(18, 10)
12
B(–4, 8)
8
4
W
E
-16 -12 -8 -4
0
S 4
8
12 16 20
If the lamp is placed at (6, 19), it will be
about the same distance from each entrance.
It will illuminate each entrance equally.
I rounded the values of x and y to the nearest integer.
Reflecting
A.
B.
NEL
Why is the intersection of two of the perpendicular bisectors
the centre of the circle that Rebecca wants?
Why did Jack only need to determine the intersection of two
of the perpendicular bisectors for the triangle?
Chapter 2
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APPLY the Math
EXAMPLE 2
Solving a problem using coordinates
The closest power line to the parking lot in Example 1 runs along a straight
line that contains points (0, 4) and (12, 10). At what point on the power
line should the cable from the lamp be connected? If each unit represents
1 m, how much cable will be needed to reach the power line? Round your
answers to the nearest tenth.
Eden’s Solution
y lamp (6, 19)
20
16
I drew a diagram. The shortest distance from the
lamp to the power line is the perpendicular
distance. I drew this on my diagram.
cable
12
(12, 10)
power line
8
4
(0, 4)
0
4
x
8
12 16
10 - 4
12 - 0
6
=
12
1
=
2
1
The equation of the power line is y = x + 4.
2
To calculate the perpendicular distance, I had to
determine the point where the perpendicular line
intersects the power line. To do this, I had to
determine the equations for the cable and the
power line.
m =
The cable is perpendicular to the power line, so the
slope of the equation for the power line is 2.
Therefore, an equation for the perpendicular line
is y = - 2x + b.
The point (6, 19) is on this line, so
19 = - 2(6) + b
19 = - 12 + b
31 = b
I determined the equation for the power line first.
I already knew that the y-intercept is 4, so I just
had to calculate the slope.
The cable from the lamp is perpendicular to the
power line. The slope of the equation for the cable
1
is the negative reciprocal of , which is 2.
2
I used this slope to write an equation for the cable.
Then I substituted the coordinates for the lamp into
the equation to determine the value of b.
The equation of the perpendicular line from (6, 19)
to the power line is y = - 2x + 31.
118
2.7 Using Coordinates to Solve Problems
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2.7
1
x + 4
2
y = - 2x + 31
y =
1
x + 4 = - 2x + 31
2
5
x = 27
2
54
x =
5
x = 10.8
I used substitution to solve the system of equations
and determine the point where the two lines
intersect.
1
(10.8) + 4
2
= 9.4
The cable from the lamp should be connected
to the power line at point (10.8, 9.4).
The corresponding value of y is y =
Length of cable 2(10.8 - 6)2 + (9.4 - 19)2
= 223.04 + 92.16
= 2115.2
#
= 10.73
I used the distance formula to calculate the length
of cable that will be needed. I rounded my answer
up to the nearest tenth of a metre to make sure
I had extra cable.
About 10.8 m of cable will be needed to connect
the lamp to the power line.
In Summary
Key Idea
• You can use the properties of lines and line segments to solve multi-step
problems when you can use coordinates for some or all of the given
information in the problem.
Need to Know
• When solving a multi-step problem, you may find it helpful to follow
these steps:
Read the problem carefully, and make sure that you understand it.
• Make a plan to solve the problem, and record your plan.
• Carry out your plan, and try to keep your work organized.
• Look over your solution, and check that your answers seem
reasonable.
• Drawing a graph and labelling it with the given information may help
you plan your solution and check your results.
• You may need to determine the coordinates of a point of intersection
before using the formulas for the slope and length of a line segment.
•
NEL
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CHECK Your Understanding
y
6
A
Questions 1 to 5 refer to the diagram at the left.
䉭ABC has vertices at A(1, 4), B(1,2), and C(5, 1). The altitude from
vertex A meets BC at point D.
2
-2
B
C x
0
D
4
1. a) Determine the slope of BC.
b) Determine the slope of AD.
c) Determine the equation of the line that contains AD.
6
2. Determine the equation of the line that contains BC.
3. Determine the coordinates of point D.
4. Determine the lengths of BC and AD.
5. Determine the area of 䉭ABC.
PRACTISING
6. A triangle has vertices at A(3, 2), B(5, 6), and C(5, 0).
a) Determine the equation of the median from vertex A.
b) Determine the equation of the altitude from vertex A.
c) Determine the equation of the perpendicular bisector of BC.
d) What type of triangle is 䉭ABC? Explain how you know.
7. Points P(9, 2) and Q(9, 2) are endpoints of a diameter of a circle.
K
a) Write the equation of the circle.
b) Show that point R(7, 6) is also on the circle.
c) Show that ∠ PRQ is a right angle.
8. 䉭LMN has vertices at L(3, 4), M(4, 3), and N(4, 1).
Use analytic geometry to determine the area of the triangle.
9. 䉭DEF has vertices at D(2, 8), E(6, 2), and F(3, 2). Use analytic
geometry to determine the coordinates of the orthocentre (the point
where the altitudes intersect).
10. 䉭PQR has vertices at P(12, 6), Q(4, 0), and R(8, 6).
Use analytic geometry to determine the coordinates of the centroid
(the point where the medians intersect).
11. 䉭JKL has vertices at J(2, 0), K(2, 8), and L(7, 3). Use analytic
geometry to determine the coordinates of the circumcentre (the point
where the perpendicular bisectors intersect).
12. A university has three student residences, which are located at points
A(2, 2), B(10, 6), and C(4, 8) on a grid. The university wants to build
a tennis court an equal distance from all three residences. Determine
the coordinates of the tennis court.
120
2.7 Using Coordinates to Solve Problems
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2.7
13. Explain two different strategies you could use to show that points
C
D, E, and F lie on the same circle, with centre C.
14. A design plan for a thin triangular computer component shows
A
the vertices at points (8, 12), (12, 4), and (2, 8). Determine
the coordinates of the centre of mass.
15. A stained glass window is in the shape of a triangle, with vertices
at A(1, 2), B(2, 1), and C(5, 0). 䉭XYZ is formed inside 䉭ABC
by joining the midpoints of the three sides. The glass that is used
for 䉭XYZ is blue, but the remainder of 䉭ABC is green. Determine
the ratio of green to blue glass used.
16. Three homes in a rural area, labelled A, B, and C in the diagram at the
right, are converting to natural gas heating. They will be connected to
the gas line labelled GH in the diagram. On a plan marked out in metres,
the coordinates of the points are A(16, 32), B(22, 24), C(56, 8),
G(16, –30), and H(38, 42).
a) Determine the length of pipe that the gas company will need to
connect the three houses to the gas line. Which homeowner will
have the highest connection charge?
b) Determine the best location for a lamp to illuminate the three
homes equally.
y
48
H
A 32
gas line
16
C x
-16
-16
G
-32
0
16 32 48 64
B
17. Determine the type of triangle that is formed by the lines x y 11,
x y 1, and x 3y 3. Justify your decision.
18. Archaeologists on a dig have found an outside fragment of an ancient
T
circular platter. They want to construct a replica of the platter for a
display. How could they use coordinates to calculate the diameter
of the platter? Include a diagram in your explanation.
19. Suppose that you know the coordinates of the vertices of a triangle.
Describe the strategy you would use to determine the equation of each
median and altitude that can be drawn from each vertex of the triangle
to the opposite side.
Extending
20. A triangle has vertices at P(1, 2), Q(4, 4), and R(1, 2). Show that
the centroid divides each median in the ratio 2:1.
Career Connection
An archaeologist searches for
clues about the lives of people
in past civilizations. Most
archaeologists are employed
by a university or a museum.
21. A circle is defined by the equation x 2 + y 2 = 10a 2.
a) Show that RQ, with endpoints R(3a, a) and Q(a, 3a), is a chord
in the circle.
b) Show that the line segment joining the centre of the circle to the
midpoint of RQ is perpendicular to RQ.
NEL
Chapter 2
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Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
• See Lesson 2.4,
Examples 1 and 2.
• Try Chapter Review
Questions 12 to 15.
Q:
How do you use the coordinates of the vertices of a triangle
or quadrilateral to determine what type of figure it is?
A:
To determine whether a triangle is isosceles, equilateral, or scalene,
you calculate the side lengths using the distance formula. To
determine whether the triangle is a right triangle, you substitute the
side lengths into the Pythagorean theorem to see if they work, or you
calculate the slopes of the line segments to see if two of the slopes are
negative reciprocals.
x
o
o o
equilateral
triangle
o
o
isosceles
triangle
scalene
triangle
x
isosceles right
triangle
right
triangle
For a quadrilateral, you determine the length and slope of each line
segment that forms a side. Then you compare the lengths to see if
there are equal sides, and compare the slopes to see if any sides are
parallel or perpendicular.
x
x
o
o
x
parallelogram
o
o
x
rectangle
rhombus
square
x
o
o
irregular
quadrilateral
Study
Aid
Examples 1 to 3.
Questions 16 to 20.
122
Chapter Review
o
o
isosceles
trapezoid
kite
Q:
How can you use the coordinates of vertices to verify
properties of triangles, quadrilaterals, or circles?
A:
You can use the coordinates of vertices to calculate midpoints and
slopes, as well as side lengths in a triangle, lengths of sides or diagonals
in a quadrilateral, or lengths of chords in a circle. Then you can use
these values to verify properties of the figure.
• See Lesson 2.5,
• Try Chapter Review
trapezoid
x
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Chapter Review
Q:
How do you use coordinates to locate a point that is
the same distance from three given points?
A:
You draw two line segments that join two pairs of given points. Then
you determine the point of intersection of the perpendicular bisectors
of these line segments. The point where the perpendicular bisectors
intersect (called the circumcentre) is the same distance from the three
given points.
Study
Aid
• See Lesson 2.7, Example 1.
• Try Chapter Review
Question 23.
circumcentre
B
A
C
Q:
How do you calculate the distance from a point to a line?
A:
The distance from a point to a line is the perpendicular distance, since
this is the shortest possible distance.
y
Study
Aid
• See Lesson 2.3, Example 3,
and Lesson 2.7, Example 2.
• Try Chapter Review
Question 25.
P(x, y)
x
0
To calculate the distance from P to the line in the diagram:
• Determine the equation of a perpendicular line that goes through P.
To do this, take the negative reciprocal of the slope of the line in the
diagram. Then use the coordinates of P to determine the y-intercept
of the perpendicular line.
• Determine the coordinates of the point of intersection of the line in
the diagram and the perpendicular line by solving the linear system
formed by the two lines.
• Determine the length of the line segment that joins P to this point
of intersection using the distance formula.
NEL
Chapter 2
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PRACTICE Questions
Lesson 2.1
10. Determine the equation of this circle.
1. On the design plan for a garden, a straight path
y
runs from (⫺25, 20) to (40, 36). A lamp is
going to be placed at the midpoint of the path.
Determine the coordinates for the lamp.
4
2
x
2. ^ABC has vertices at A(⫺4, 4), B(⫺4, ⫺2),
and C(2, ⫺2).
a) Determine the equation of the median
from B to AC.
b) Is the median for part a) also an altitude?
Explain how you know.
3. ^LMN has vertices at L(0, 4), M(⫺5, 2), and
N(2, ⫺2). Determine the equation of the
perpendicular bisector that passes through MN.
Lesson 2.2
-4
2
-2
4
-4
11. The point (⫺2, k) lies on the circle
x 2 + y 2 = 20. Determine the values of k.
Show all the steps in your solution.
Lesson 2.4
12. ^ABC has vertices as shown. Use analytic
geometry to show that ^ABC is isosceles.
4. Which point is closer to the origin: P(⫺24, 56)
8
or Q(35, ⫺43)?
house to a temporary power supply. On the
plan, the coordinates of the house are (20, 110)
and the coordinates of the power supply are
(105, 82). What is the least amount of cable
needed?
6. ^QRS has vertices at Q(2, 6), R(⫺3, 1), and
S(6, 2). Determine the perimeter of the triangle.
7. ^XYZ has vertices at X(1, 6), Y(⫺3, 2), and
Z(9, 4). Determine the length of the longest
median in the triangle.
Lesson 2.3
8. a) Determine the equation of the circle that is
centred at (0, 0) and passes through point
(⫺8, 15).
b) Identify the coordinates of the intercepts
and three other points on the circle.
9. A circle has a diameter with endpoints
C(20, ⫺21) and D(⫺20, 21). Determine
the equation of the circle.
Chapter Review
y
6
5. A builder needs to connect a partially built
124
-2
0
C
B
4
2
x
A
-2
0
-2
2
4
6
8
13. A triangle has vertices at A(1, 1), B(–2, –1), and
C(3, ⫺2). Calculate the side lengths to
determine whether the triangle is isosceles,
equilateral, or scalene.
14. Show that the quadrilateral with vertices
at J(⫺1, 1), K(3, 4), L(8, 4), and M(4, 1)
is a rhombus.
15. Determine the type of quadrilateral described by
the vertices R(⫺3, 2), S(⫺1, 6), T(3, 5), and
U(1, 1). Show all the steps in your solution.
Lesson 2.5
16. A quadrilateral has vertices at A(–3, 1),
B(–5, ⫺9), C(7, ⫺1), and D(3, 3). Show that
the midsegments of the quadrilateral form
a parallelogram.
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Chapter Review
17. Show that points (10, 10), (⫺7, 3), and
(0, ⫺14) lie on a circle with centre (5, ⫺2).
18. A triangle has vertices at P(⫺2, 7), Q(⫺4, 2),
and R(6, ⫺2).
a) Show that ^PQR is a right triangle.
b) Show that the midpoint of the hypotenuse is
the same distance from each vertex.
22. ^XYZ has vertices at X(0, 1), Y(6, ⫺1), and
Z(3, 6). Use analytic geometry to determine
the coordinates of the centroid (the point
where the medians intersect).
y
4
19. a) Show that points (6, 7) and (⫺9, 2) are the
endpoints of a chord in a circle with centre
(0, 0).
b) A line is drawn through the centre of the
circle so that it is perpendicular to the
chord. Verify that this line passes through
the midpoint of the chord.
20. a) Quadrilateral JKLM has vertices as shown.
Show that the diagonals of the quadrilateral
bisect each other.
Z
6
2
X
-2
0
-2
x
2
4
Y 8
23. A new lookout tower is going to be built so that
it is the same distance from three ranger stations.
If the stations are at A(⫺90, 28), B(0, ⫺35),
and C(125, 20) on a grid, determine the
coordinates of the point where the new tower
should be built.
y
2
x
-2
0
2
-2
4 J 6
8
K
-4
M
-6
L
-8
b) Make a conjecture about the type of
quadrilateral JKLM could be.
c) Use analytic geometry to verify your
conjecture.
Lesson 2.7
21. ^PQR has vertices at P(0, ⫺2), Q(4, 4), and
R(⫺4, 5). Use analytic geometry to determine
the coordinates of the orthocentre (the point
where the altitudes intersect).
y
R
Q
4
2
x
-4 -2
NEL
0
-2 P
2
4
24. Predict the type of quadrilateral that is formed
by the points of intersection of the lines
3x + y - 4 = 0, 4x - 5y + 30 = 0,
y = - 3x - 1, and -4x + 5y + 10 = 0. Give
reasons for your prediction. Verify that your
prediction is correct by solving this problem.
25. A builder wants to run a temporary line from
the main power line to a point near his site
office. On the site plan, the site office is at
S(25, 18) and the main power line goes through
points T(1, 5) and U(29, 12). Each unit
represents 1 m.
a) At what point should the builder connect
to the main power line?
b) What length of cable will the builder need?
Chapter 2
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Chapter Self-Test
1. An underground cable is going to be laid between points A(⫺6, 23)
Checklist
✔ Questions 1, 2, 6, and 8:
Did you make connections
between analytical
geometry and the
situation?
✔ Questions 3 and 4: Did you
apply reasoning skills to
construct a mathematical
argument to confirm each
figure type?
and B(14, ⫺12).
a) If each unit represents 1 m, what length of cable will be needed?
Give your answer to the nearest metre.
b) An access point will be located halfway between the endpoints of
the cable. At what coordinates should the access point be built?
2. A stone is tossed into a pond, creating a circular ripple. The radius
of the ripple increases by 12 cm/s.
✔ Question 7: Did you use
appropriate mathematical
vocabulary to communicate
your thinking?
a) Write an equation that describes the ripple exactly 3 s after
the stone lands in the water. Use the origin as the point
where the stone lands in the water.
b) A bulrush is located at point (⫺36, 48). When will the ripple
reach the bulrush?
3. The triangle at the left has vertices at A(1, 2), B(⫺3, ⫺1), and
y
A
2
x
B -2
0
2
-2
4
-4
-6
C
C(0, ⫺5). Use analytic geometry to show that the triangle is an isosceles
right triangle.
4. The corners of a building lot are marked at P(⫺39, 39), Q(⫺78, ⫺13),
R(26, ⫺91), and S(65, ⫺39) on a grid.
a) Verify that PQRS is a rectangle.
b) What is the perimeter of the building?
5. Quadrilateral JKLM has vertices at J(2, 4), K(6, 1), L(2, ⫺2), and
M(⫺2, 1). What type of quadrilateral is JKLM?
6. Three straight paths in a park form a triangle with vertices
at A(⫺24, 16), B(56, ⫺16), and C(⫺72, ⫺32). A new fountain
is the same distance from the intersections of the three paths.
Determine the location of the new fountain.
7. Explain how you can use analytic geometry to calculate the distance from
a known point to a line that passes through two other known points.
8. The sides of a triangle are defined by the equations x + 2y - 2 = 0,
2x - y - 4 = 0, and 3x + y + 9 = 0. Determine the type
of triangle that is formed by these three sides.
126
Chapter Self-Test
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Chapter Task
X Marks the Spot
A new diagnostic centre, with laboratories and computer-imaging
equipment, is being planned. The centre will serve four walk-in clinics.
Safety Connection
A protective cabinet, as well as
protective gloves and jacket,
are needed for handling
materials in a laboratory.
On a map, these clinics are located at A(1, 12), B(12, 19), C(19, 8),
and D(3, 2). On the map, 1 unit represents 1 km.
?
Where should the diagnostic centre be located?
A.
On a grid, show the locations of the walk-in clinics. Estimate where
you think the diagnostic centre should go, and mark this point.
B.
Use coordinates to determine a point that is the same distance
from the clinics at points A, B, and C.
C.
Calculate the distance from the clinic at point D to the point
you determined for part B.
D.
E.
Checklist
✔ Did you show all your
steps?
Use coordinates to determine a point that is the same distance
from the clinics at points A, B, and D.
✔ Did you draw and label
Calculate the distance from the clinic at point C to the point
you determined for part D.
✔ Did you support your
F.
Repeat parts D and E for the other combinations of three clinics.
G.
Based on your results, choose the best location for the diagnostic
centre. Justify your choice.
NEL
Task
your diagram accurately?
choice of location for the
diagnostic centre?
✔ Did you explain your
thinking clearly?
Chapter 2
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Altitude (m)
2000
5/12/09
1500
1000
500
Altitude (m)
2000
x
5 10 15 20
Time (min)
y
1500
1000
500
x
0
2000
Altitude (m)
Page 128
y
0
5 10 15
Time (min)
y
1500
1000
500
0
128
10:43 AM
x
5 10 15
Time (min)
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Chapter
3
Graphs of
Quadratic
Relations
GOALS
You will be able to
• Describe the graphs and properties
of quadratic relations of the forms
y ⴝ ax2 ⴙ bx ⴙ c and y ⴝ a(x ⴚ r)(x ⴚ s)
• Expand and simplify quadratic
expressions
• Apply quadratic models to solve
problems
? What story does each graph
tell about the movement of
these balloons?
NEL
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Getting Started
WORDS YOU NEED to Know
1. Match each term with the correct diagram or example.
a) linear relation
b) first differences
iii)
ii)
iv)
x
y
1
12
2
14
3
16
14 ⫺ 12 ⫽ 2
2
2
4
18
5
20
2
e) intercepts
f ) line of symmetry
v) y = 3x + 5
Value of tractor ($)
i) a(b + c) = ab + ac
c) distributive property
d) scatter plot
45 000
Value of a Tractor
y
vi)
-2
40 000
35 000
-1
(2, 0)
0
-2
-4
30 000
1
2
x
3
(0, -4)
-6
25 000
0
y
2
-8
x
2 4 6 8
Age of tractor (years)
SKILLS AND CONCEPTS You Need
Curves as Mathematical Models
Aid
• For more help and practice,
see Appendix A-11.
Age
(years)
Resting Heart
Rate (beats
per minute)
21
60
24
61
26
63
29
65
31
68
35
73
39
78
130
Getting Started
Sometimes, a curve is the best model for the relationship between the
dependent variable and independent variable in a relation.
EXAMPLE
The resting heart rates of people of different ages are listed in the table.
Estimate the age of a person with a resting heart rate of 70 beats per minute.
Solution
Heart Rate
Resting heart rate
(beats per minute)
Study
90
y
80
70
60
50
0
x
10 20 30 40 50
Age (years)
Create a scatter plot, and draw
a curve. Interpolate by drawing
a horizontal line from 70 on
the Resting heart rate axis until
it touches the curve. Draw
a vertical line down to the Age
axis. A person with a resting
heart rate of 70 beats per
minute is estimated to be about
34 years old.
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Getting Started
2. Use the graph for resting heart rates to estimate
a) the resting heart rate of a person who is 15 years old
b) the age of a person with a resting heart rate of 75
3. This table shows the height of a baseball after it has been hit.
Time (s)
0
1
2
3
4
5
Height (m)
0.5
20.5
30.5
30.5
20.5
0.5
a) Create a scatter plot and draw a curve.
b) Estimate the height of the baseball at 2.5 s.
c) Estimate when the baseball will have a height of 25 m.
Multiplying a Polynomial by a Monomial
Several strategies can be used to multiply a polynomial by a monomial.
Study
EXAMPLE
• For more help and practice,
Aid
see Appendix A-8.
Multiply 2x(3x + 5).
Solution
Using an Algebra Tile Model
x
x
x
1 1 1 1 1
x
x2
x2
x2
x x x x x
x
x2
x2
x2
x x x x x
Using the Distributive Property
2x(3x + 5) = 2x(3x) + 2x(5)
= 6x 2 + 10x
2x(3x + 5) = 6x 2 + 10x
4. Expand and simplify each expression.
a) 4(x + 3)
b) 2x(x - 5)
c) - 3x 2(x - 2)
d) 4x(2x - 3) + 3x(7 - 5x)
e) 7x 2(4x - 7 + 2x 2) - x(3x 2 - 5x - 2)
f ) - 4x(x 3 - 3x 2) + 2x(5x 2 - 3x) - 6x 3(x - 3)
NEL
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PRACTICE
Aid
Study
10:44 AM
5. Determine the value of y for each given value of x.
• For help, see the Review
of Essential Skills and
Knowledge Appendix.
c) y = x 2 + 2x - 1; x = 4
d) y = (2x + 1)(x - 3); x = 2
a) y = 2x - 3; x = 1.5
b) y = x 2; x = - 3
Question
Appendix
5
A-5
6 to 9
A-7
6. A cell-phone plan costs $25 each month plus $0.10 per minute of
airtime. Make a table of values, construct a graph, and write an
equation to represent the monthly cost of the plan.
7. A laptop was purchased new for $1000 and depreciates by $200 each
year. Make a table of values, construct a graph, and write an equation
to represent the value of the laptop.
8. State the x- and y-intercepts for the relation in the graph at the left.
y
2
x
-8 -6 -4
-2
0
-2
2
-4
4
9. Determine the x- and y-intercepts for each relation. Then sketch
the graph.
a) y = 2x - 3
b) x + y = 5
c) y = 2x + 5
d) 3x + 2y = 12
e) x = 2
f) y = -5
-6
10. State whether each statement is true or false. If the statement is false,
-8
create an example to support your decision.
a) Every table of values for a relation has constant first differences.
b) Every table of values for a linear relation has first differences
equal to 0.
c) A relation can have no more than one y-intercept.
d) A relation can have no more than one x-intercept.
e) All 2-D figures have a line of symmetry.
-10
11. Copy and complete the chart to show what you know about
nonlinear relations.
Definition:
Examples:
C ⫽ ␲r2
Characteristics:
Nonlinear
Relations
8
Non-examples:
y
x 0 1 2 3 4
y 1 3 9 27 81 6
4
2
0
132
Getting Started
x
2
4
6
8
10
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Getting Started
APPLYING What You Know
YOU WILL NEED
Analyzing Balloon Data
• grid paper
• ruler
Kajsa is a member of a ballooning club. She is responsible for analyzing the
data collected by the club’s training school on flights piloted by students.
The following tables show data for two different training flights.
Training Flight 1
Time (s)
0
1
2
3
4
5
6
Height (m)
270
260
250
240
230
220
210
Time (s)
0
1
2
3
4
5
6
Height (m)
20
33
48
50
64
76
90
Training Flight 2
?
How can you tell whether the data sets for the training flights
are linear?
A.
In which training flight is the balloon rising? In which training flight
is the balloon descending? Explain how you know.
B.
Describe three different strategies you could use to determine whether
a data set is linear.
C.
Use each strategy you described for part B to determine whether
the data sets for the training flights are linear.
NEL
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Exploring Quadratic Relations
YOU WILL NEED
GOAL
• graphing calculator
Determine the properties of quadratic relations.
EXPLORE the Math
24
Height (m)
20
y
y = 5x2 20x 1
16
12
8
4
0
x
1
2
3 4
Time (s)
5
quadratic relation in standard
form
a relation of the form
y ax2 bx c, where
a Z 0; for example,
y 3x2 4x 2
Tech
A “pop fly” in baseball occurs when
the ball is hit straight up by the
batter. For a certain pop fly, the
height of the ball above the ground,
y, in metres, is modelled by the
relation y = - 5x 2 + 20x + 1,
where x is the time in seconds after
the ball leaves the bat.
The path of the ball is straight up
and down. The graph of the relation does not show this, however, because
the ball rises fast at first and then more slowly due to gravity. The relation
for the height of the baseball is an example of a quadratic relation in
standard form.
?
How does changing the coefficients and constant in
y ax2 bx c affect the graph of the quadratic relation?
A.
Enter y = x 2 into a graphing calculator. Scroll to the left of Y1, and
press ENTER to activate a thick line, then graph the relation by
pressing GRAPH. Use the window settings shown.
Support
For help using a TI-83/84
graphing calculator to graph
relations and create difference
tables, see Appendix B-2 and
B-7. If you are using a TI-nspire,
see Appendix B-38 and B-43.
Describe the shape of the graph and its symmetry.
134
B.
Create a table of values for x from 4 to 4.
C.
Calculate the first differences of the y-values. What do they confirm
about the graph of the relation?
3.1 Exploring Quadratic Relations
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3.1
D.
Calculate the second differences. What do you notice?
E.
Investigate other relations of the form y = ax 2, where a 7 0, b = 0,
and c = 0. Repeat parts B to D for each relation in the graphing
calculator screen at the right. Describe how the graphs and difference
tables for these relations are the same and how they are different.
F.
Investigate relations of the form y = ax 2, where a 6 0, b = 0, and
c = 0. Repeat parts B to D using these relations for Y2 to Y6:
y = - x 2, y = - 2x 2, y = - 5x 2, y = - 0.5x 2, and y = - 0.2x 2.
Describe how the parabolas and difference tables are the same and
how they are different.
G.
Investigate relations of the form y = ax 2 + c , where b = 0. Repeat
parts B to D using these relations for Y2 to Y5: y = x 2 + 2,
y = x 2 - 4, y = x 2 + 5, and y = x 2 - 6. Make a conjecture
to describe how changing the value of c affects a parabola.
H.
Test your conjecture for part G by entering five new equations with
the same value of a, but different values of c.
I.
Investigate relations of the form y = ax 2 + bx + c . Repeat parts B
to D using these relations for Y2 to Y5: y = x 2 + 2x + 2,
y = x 2 - 4x + 2, y = x 2 + 5x + 2, and y = x 2 - 6x + 2. Make
a conjecture to describe how changing the value of b affects a parabola.
J.
Test your conjecture for part I by entering five new equations with the
same value of a and the same value of c, but different values of b.
second differences
values that are calculated by
subtracting consecutive first
differences in a table of values
parabola
a symmetric graph of a quadratic
relation, shaped like the letter
“U” right-side up or upside
down
Reflecting
K.
Describe what you noticed in the quadratic relations
you examined about
i) the degree of each equation
ii) the second differences in the tables of values
L.
Explain how the value of a in y = ax 2 + bx + c relates to
i) the graph of the parabola
ii) the second differences in the table of values
M. Explain whether changing the value of b or c changes
i) the location of the y-intercept of a parabola
ii) the location of the line of symmetry of a parabola
NEL
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In Summary
Key Ideas
• The graph of any quadratic relation of the form y ax2 bx c,
where a Z 0, is a parabola that has a vertical line of symmetry.
• Any relation described by a polynomial of degree 2 is quadratic.
Need to Know
y ax2 bx c
y ax2 bx c
• For the quadratic relation
2
y ax bx c,
a0
a0
y
y
• the second differences are
constant, but not zero
x
x
• when the value of a (the
2
0
0
coefficient of the x term) is
positive, the parabola opens
upward and the second
differences are positive
• when the value of a (the coefficient of the x2 term) is negative, the
parabola opens downward and the second differences are negative
• changing the value of b (the coefficient of the x term) changes
the location of the line of symmetry of the parabola
• the constant c is the value of the y-intercept of the parabola
FURTHER Your Understanding
1. Which graphs appear to represent a quadratic relation? Explain.
a)
c)
y
x
x
x
0
b)
y
e)
y
0
0
d)
y
x
y
f)
y
x
x
0
0
0
136
3.1 Exploring Quadratic Relations
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3.1
2. a) Determine the degree of each relation.
i) y = 5x - 2
ii) y = x 2 - 6x + 4
iii) y = x(x + 4)
iv) y = 2x 3 - 4x 2 + 5x - 1
b) Which relations in part a) have a graph that is a parabola?
3. State the y-intercept of each quadratic relation in question 2.
4. Calculate the first differences for each set of data, and determine
whether the relation is linear or nonlinear. If the relation is nonlinear,
determine the second differences and identify the quadratic relations.
a)
b)
c)
x
10
20
30
40
y
21
41
61
81
x
1
2
3
4
y
4
7
12
17
x
5
6
7
8
y
2
3
5
8
d)
e)
f)
x
0
1
2
3
y
1
1
7
11
x
0
1
2
3
y
2
1
6
25
x
0
1
2
3
4
y
1
2
4
8
16
5. Each table of values represents a quadratic relation. Decide, without
graphing, whether the parabola opens upward or downward.
a)
x 3
y
b)
2
2.5
5.0
c)
1
0
6.5
7.0
x 2
1
0
1
2
y
5
0
15
40
0
d)
x
2
1
0
1
2
y
3
3
5
3
3
x
0
1
2
3
4
y
1
4
15
32
55
6. State whether the graph of each quadratic relation opens upward or
downward. Explain how you know.
a) y = x 2 - 1
b) y = - x 2 + 5x
1
c) y = - x 2 + 6x - 4
2
1
d) y = - 9x + x 2 + 6
2
7. Explain why the condition a Z 0 must be stated to ensure that
y = ax 2 + bx + c is a quadratic relation.
NEL
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Properties of Graphs of
Quadratic Relations
YOU WILL NEED
GOAL
• grid paper
• ruler
• graphing calculator
Describe the key features of the graphs of quadratic relations,
and use the graphs to solve problems.
LEARN ABOUT the Math
Grace hits a golf ball out of a sand trap, from a position that is level
with the green. The path of the ball is approximated by the equation
y = - x 2 + 5x, where x represents the horizontal distance travelled by
the ball in metres and y represents the height of the ball in metres.
?
What is the greatest height reached by the ball and how far
away does it land?
y
x
EXAMPLE 1
Reasoning from a table of values and
a graph of a quadratic model
Determine the greatest height of the ball and the distance away that
it lands.
Health Connection
Ultraviolet sun rays can
damage the skin and cause
skin cancer. Wearing a hat
with a broad brim around the
entire hat provides protection.
Erika’s Solution
x
0
1
2
3
4
5
y
0
4
6
6
4
0
Height (m)
8
y
6
4
2
0
x
2 4 6 8
Horizontal distance (m)
138
I made a table of values. I used
only positive values of x since the
ball moves forward, not
backward, when hit. When
I reached a y-value of 0,
I stopped. I assumed that the ball
would not go below ground level.
3.2 Properties of Graphs of Quadratic Relations
I used my table of values to
sketch the graph. I knew that the
graph was a parabola, since the
degree of the equation is 2.
The parabola has a vertical line
of symmetry that appears to
pass through x 2.5.
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3.2
0 + 5
= 2.5.
2
1 + 4
When y = 4, x =
= 2.5.
2
2 + 3
When y = 6, x =
= 2.5.
2
The equation of the axis of symmetry
is x = 2.5.
I noticed that the points on
the parabola with the same
y-coordinate were the same
distance from the line of
symmetry. I reasoned that the
axis of symmetry is the
perpendicular bisector of any line
segment joining points with the
same y-coordinates. The means
of the x-coordinates of these
points give the equation of the
axis of symmetry.
y = - x 2 + 5x
y = - (2.5)2 + 5(2.5)
y = - 6.25 + 12.5
y = 6.25
The coordinates of the vertex
are (2.5, 6.25).
I saw that the vertex intersects
the axis of symmetry, so its
x-coordinate is 2.5. I substituted
this value of x into the equation
to get the maximum value.
When y = 0, x =
Height (m)
8
y
6
a line that separates a 2-D figure
into two identical parts; if the
figure is folded along this line,
one of these parts fits exactly on
the other part
vertex
the point of intersection
of a parabola and its axis
of symmetry
maximum value
the greatest value of the
dependent variable in a relation
x 2.5
(2.5, 6.25)
4
2
0
axis of symmetry
2.5 m
(5, 0) x
From my graph, I saw that y 0
when x 5. So, the ball lands
5 m away from where it was hit.
2 4 6 8
Horizontal distance (m)
The ball’s greatest height is 6.25 m.
This occurs at a horizontal distance of
2.5 m from the starting point. The
ball lands 5 m from the starting point.
Reflecting
A.
Was the table of values or the graph more useful for determining the
maximum height of the ball and the distance between where it was hit
and where it landed? Explain.
B.
How is the x-value at the maximum height of the ball related to
the x-value of the point where the ball touches the ground?
C.
Is it possible to predict whether a quadratic relation has a maximum
value if you know the equation of the relation? Explain.
NEL
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APPLY the Math
EXAMPLE 2
Selecting a table of values strategy to graph a quadratic relation
Sketch the graph of the relation y = x 2 - 6x. Determine the equation
of the axis of symmetry, the coordinates of the vertex, the y-intercept, and
the x-intercepts.
Cassandra’s Solution
The relation y = x 2 - 6x is quadratic.
a = 1, b = - 6, and c = 0
x
y
10
1
7
8
0
0
6
1
5
4
2
8
2
3
9
4
8
5
5
-4
6
0
-6
7
7
-8
The degree of the equation is 2, so the graph is a
parabola. The coefficient of the x2 term is a 1.
Since a is positive, the parabola opens upward.
y
y x2 6x
x
-4 -2
0
-2
2
4
6
8
10
I created a table of values using some negative and
some positive x-values. I plotted each ordered pair,
and drew a parabola that passed through each
point. The parabola appears to have (3, 9) as its
vertex and x 3 as its axis of symmetry.
-10
The points (1, - 5) and (5, - 5) are directly
across from each other on the parabola.
1 + 5
x =
so x = 3
2
The equation of the axis of symmetry is x = 3.
When x = 3 in y = x 2 - 6x,
y = 32 - 6(3)
y = 9 - 18
y = -9
The vertex occurs at (3, - 9).
140
3.2 Properties of Graphs of Quadratic Relations
I verified the equation of the axis of symmetry
by averaging the x-coordinates of two points with
the same y-value.
Since the vertex is on the axis of symmetry and the
parabola, I substituted x 3 into y x2 6x.
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3.2
y
10
8
6
x-intercepts
(0, 0) and (6, 0)
4
y-intercept
(0, 0) 2
-4 -2
x
0
-2
-4
2
4
6
8
I looked at my graph to determine its features.
10
equation of the axis of
symmetry x 3
-6
y x2 6x
-8
-10
vertex (3, 9)
This parabola has x = 3 as the equation of its axis
of symmetrry, the vertex is located at (3, -9), the
y-intercept is 0, and the x-intercepts are 0 and 6.
EXAMPLE 3
Selecting a strategy to determine
the minimum value
A kingfisher dives into a lake. The underwater path of the bird is described
by a parabola with the equation y = 0.5x 2 - 3x, where x is the horizontal
position of the bird relative to its entry point and y is the depth of the bird
underwater. Both measurements are in metres.
Graph the parabola. Use your graph to determine the equation of the axis
of symmetry, the coordinates of the vertex, the y-intercept, and the
x-intercepts. Calculate the bird’s greatest depth below the water surface.
Environment Connection
Pauline’s Solution
Since a 0.5, the parabola opens
upward. The deepest point of the
kingfisher’s path is the minimum value
of the relation. This occurs at the vertex
of the parabola and corresponds to the
y-coordinate of the vertex.
NEL
I made a plan to solve
the problem.
Since the Belted Kingfisher eats
fish and crayfish, it is at risk
due to toxins such as mercury.
minimum value
the least value of the dependent
variable in a relation
Chapter 3
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x
y
0
0.0
1
2.5
2
4.0
3
4.5
4
4.0
5
2.5
6
0.0
Page 142
I created a table of values using
the equation. I assumed that the
bird moved from left to right, so
I chose only positive values of x.
y
2
x
0
-2
2
4
6
8
-4
-6
I plotted the points to create a
graph; the vertex of the parabola
appears to be (3, 4.5) and the
equation of the axis of symmetry
appears to be x 3.
The y-intercept occurs at (0, 0).
The x-intercepts occur at (0, 0)
and (6, 0).
I looked at my graph to
determine the intercepts.
0 + 6
2
The equation of the axis of
symmetry is x = 3.
The axis of symmetry is halfway
between the x-intercepts, so
I calculated the mean of the
x-coordinates.
y = 0.5x 2 - 3x
y = 0.5(3)2 - 3(3)
y = 4.5 - 9
y = - 4.5
The vertex is on the axis of
symmetry, so I substituted x 3
into the equation to determine
the y-coordinate.
x =
The vertex is (3, -4.5).
The greatest depth of the bird, below
the surface of the water, is 4.5 m.
Tech
Support
For help graphing a relation
and determining its minimum
value using a TI-83/84 graphing
calculator, see Appendix B-9.
If you are using a TI-nspire,
see Appendix B-45.
142
3.2 Properties of Graphs of Quadratic Relations
I verified the minimum value
of the relation using the
minimum operation on
a graphing calculator.
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3.2
EXAMPLE 4
Connecting a situation to a quadratic
model
A model rocket is shot into the air from the roof of a building. Its height,
h, above the ground, measured in metres, can be modelled by the equation
h = - 5t 2 + 35t + 5, where t is the time elapsed since liftoff in seconds.
a) Determine the greatest height reached by the rocket.
b) How long is the rocket in flight?
c) Determine the height of the building.
d) When is the height of the rocket 61.25 m?
Liam’s Solution
a) The equation is quadratic. Since
a = - 5, the graph is a parabola
that opens downward. The
greatest height occurs at the
vertex. I entered the equation
y = - 5x 2 + 35x + 5 into
a graphing calculator.
Since the calculator uses the
variables x and y, I replaced the
dependent variable h with y and
the independent variable t with x.
I adjusted the window settings
until I could see the vertex.
I used the maximum operation
to determine the coordinates.
Tech
Support
For help graphing a relation,
determining its maximum
value, and determining its
x-intercepts using a TI-83/84
graphing calculator, see
Appendix B-2, B-9, and B-8.
If you are using a TI-nspire, see
Appendix B-38, B-45, and B-44.
The vertex is (3.5, 66.25).
At 3.5 s after liftoff, the rocket reaches
its greatest height of 66.25 m.
b)
The second zero (or x-intercept)
corresponds to the rocket hitting
the ground. I used the zero
operation to determine the
coordinates of this point.
Communication
Tip
The zeros of a relation are its
x-intercepts. “Zero” is another
name for “x-intercept.”
The rocket is in flight for about 7.14 s.
NEL
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c) Let x = 0.
y = - 5(0)2 + 35(0) + 5
y = 0 + 0 + 5
y = 5
Tech
Support
The height of the building
corresponds to the y-intercept of
the graph. This is the initial height
of the ball, so I substituted x 0
into the equation and solved for y.
I verified my answer on a
graphing calculator, using the
value operation.
For help determining the value
of a relation using a TI-83/84
graphing calculator, see
Appendix B-3. If you are using
a TI-nspire, see Appendix B-39.
The building is 5.00 m tall.
d)
I had to determine when the
height is 61.25 m. I entered the
relation y 61.25 into Y2 of the
equation editor and re-graphed.
The x-coordinates of the points of
intersection of the horizontal line
and the parabola tell me when
this height occurs.
Tech
Support
For help determining the points
of intersection for two relations
using a TI-83/84 graphing
calculator, see Appendix B-11.
If you are using a TI-nspire,
see Appendix B-47.
I used the intersect operation to
determine the points of
intersection. The first coordinate
of each point represents a time
when the rocket is 61.25 m
above the ground.
The rocket reaches a height of 61.25 m
after 2.5 s on the way up and after 4.5 s
on the way down.
144
3.2 Properties of Graphs of Quadratic Relations
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3.2
In Summary
Key Ideas
• The vertex of a parabola with equation y ax2 bx c is the point
on the graph with
• the least y-coordinate, or minimum value, if the parabola opens upward
• the greatest y-coordinate, or maximum value, if the parabola opens
downward
y = ax2 bx c, a 0
y
vertex
y = ax2 bx c, a 0
y
(maximum value)
vertex
(minimum value) x
x
0
0
axis of
symmetry
axis of
symmetry
• A parabola with equation y ax2 bx c is symmetrical with respect
to a vertical line through its vertex. This line, or axis of symmetry, is the
perpendicular bisector of any line segment that joins two points with
the same y-coordinate on the parabola.
Need to Know
• The x-intercepts, or zeros, of a parabola can be determined by setting
y 0 in the equation of the parabola and solving for x.
• The y-intercept of a parabola can be determined by setting x 0
in the equation of the parabola and solving for y.
• When a problem can be modelled by a quadratic relation, the graph
of the relation can be used to estimate solutions to the problem.
CHECK Your Understanding
1. For each graph, state the y-intercept, the zeros, the coordinates
of the vertex, and the equation of the axis of symmetry.
a)
b)
y
c)
y
4
2
4
2
2
x
-6 -4 -2
0
-2
-4
NEL
y
4
2
x
-4 -2
0
-2
-4
2
4
x
-2
0
-2
2
4
6
-4
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2. State the maximum or minimum value of each relation in question 1.
3. Two parabolas have the same x-intercepts, at (0, 0) and (10, 0). One
parabola has a maximum value of 2. The other parabola has a minimum
value of 4. Sketch the graphs of the parabolas on the same axes.
PRACTISING
4. Examine each parabola.
i) Determine the coordinates of the vertex.
ii) Determine the zeros.
iii) Determine the equation of the axis of symmetry.
iv) If you calculated the second differences, would they be positive or
negative? Explain.
y
a)
y
b)
4
4
2
2
x
-2
0
2
-2
4
x
6
-2
-4
0
-2
2
4
6
-4
5. The zeros of a quadratic relation occur at x = 0 and x = 6.
The second differences are positive.
a) Is the y-value of the vertex a maximum value or a minimum
value? Explain.
b) Is the y-value of the vertex a positive number or a negative
number? Explain.
c) Determine the x-value of the vertex.
6. For each quadratic relation, state
i) the equation of the axis of symmetry
ii) the coordinates of the vertex
iii) the y-intercept
iv) the zeros
v) the maximum or minimum value
y
a)
b)
4
4
x
-12 -8 -4
0
-4
4
-8
146
3.2 Properties of Graphs of Quadratic Relations
y
c)
8
x
-8 -4
0
-4
-8
4
y
4
8
x
-8 -4
0
-4
4
8
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3.2
7. Create a table of values for each quadratic relation, and sketch
its graph. Then determine
i) the equation of the axis of symmetry
ii) the coordinates of the vertex
iii) the y-intercept
iv) the zeros
v) the maximum or minimum value
a) y = x 2 + 2
b) y = - x 2 - 1
c) y = x 2 - 2x
d) y = - x 2 + 4x
e) y = x 2 - 2x + 1
f ) y = - x 2 - 2x + 3
8. Use technology to graph each quadratic relation below. Then determine
a) y = x 2 - 4x + 3
b) y = - x 2 + 4
c) y = x 2 + 6x + 8
d) y = - x 2 + 6x - 5
e) y = 2x(x - 4)
f ) y = - 0.5x(x - 8)
9. Each pair of points is located on opposite sides of the same parabola.
Determine the equation of the axis of symmetry for each parabola.
a) (3, 2), (9, 2)
c) (5.25, 2.5), (3.75, 2.5)
1
1
b) (–18, 3), (7, 3)
d) a -4 , 5 b, a -1 , 5 b
2
2
i) the equation of the axis of symmetry
ii) the coordinates of the vertex
iii) the y-intercept
iv) the zeros
v) the maximum or minimum value
10. Jen knows that (1, 41) and (5, 41) lie on a parabola defined by the
K
equation y = 4x 2 - 16x + 21. What are the coordinates of the vertex?
11. State whether you agree or disagree with each statement. Explain why.
C
a) All quadratic relations of the form y = ax 2 + bx + c have
two zeros.
b) All quadratic relations of the form y = ax 2 + bx + c have
one y-intercept.
c) All parabolas that open downward have second differences that
are positive.
Use a graphing calculator to answer questions 12 to 15.
12. A football is kicked into the air. Its height above the ground is
approximated by the relation h = 20t - 5t 2, where h is the height
in metres and t is the time in seconds since the football was kicked.
a) What are the zeros of the relation? When does the football hit
the ground?
b) What are the coordinates of the vertex?
c) Use the information you found for parts a) and b) to graph
the relation.
d) What is the maximum height reached by the football? After how
many seconds does the maximum height occur?
NEL
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13. A company that manufactures MP3 players uses the relation
A
P = 120x - 60x 2 to model its profit. The variable x represents the
number of thousands of MP3 players sold. The variable P represents
the profit in thousands of dollars.
a) What is the maximum profit the company can earn?
b) How many MP3 players must be sold to earn this profit?
c) The company “breaks even” when the profit is zero. Are there any
break-even points for this company? If so, how many MP3 players
are sold at the break-even points?
14. An inflatable raft is dropped from a hovering helicopter to a boat
in distress below. The height of the raft above the water, in metres,
is approximated by the equation y = 500 - 5x 2, where x is
the time in seconds since the raft was dropped.
a) What is the height of the helicopter above the water?
b) When does the raft reach the water?
c) What is the height of the raft above the water 6 s after
it is dropped?
d) When is the raft 100 m above the water?
Career Connection
Coast guard rescuers drop
rafts that inflate within
seconds to keep people afloat.
15. Gamez Inc. makes handheld video game players. Last year,
T
accountants modelled the company’s profit using the equation
P = - 5x 2 + 60x - 135. This year, accountants used the equation
P = - 7x 2 + 70x - 63. In both equations, P is the profit, in
hundreds of thousands of dollars, and x is the number of game players
sold, in hundreds of thousands. If the same number of game players were
sold in these years, did Gamez Inc.’s profit increase? Justify your answer.
16. a) Explain how the value of a in a quadratic relation, given in
standard form, can be used to determine if the quadratic relation
has a maximum value or a minimum value.
b) Explain how the coordinates of the vertex are related to the
maximum or minimum value of the parabola.
Extending
17. a) Determine first and second differences for each relation.
i) x = 2y 2
iii) y = x 3
x
ii) y = 2
iv) y = 2x 4
b) Are graphs for any of the relations in part a) parabolas? Explain.
c) Are any of the relations in part a) quadratic? Explain.
18. The x-coordinate of the vertex of the graph of y = 5x 2 - 3.2x + 8
is x = 0.32. The number 0.32 is very similar to 3.2, which is the
coefficient of x in the equation. Is this just a coincidence? Investigate
several examples. Then make a conjecture and try to prove it.
148
3.2 Properties of Graphs of Quadratic Relations
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Curious Math
Curious Math
Folding Paper to Create a Parabola
People in early civilizations knew that the parabola was an important
curve. In 350 BCE, Menaechmus, a student of Plato and Eudoxus, studied
the curves that are formed when a plane intersects the surface of a cone.
He discovered that there are several possibilities and that one of these
possibilities is a parabola.
In 220 BCE, Apollonius named the curves
shown. In 212 BCE, Archimedes studied the
properties of these curves. One of the properties
can be used to create a parabola by folding paper.
YOU WILL NEED
• uncreased paper or waxed
paper
• dark marker
• ruler
parabola
1. Begin with a clean, uncreased piece of paper.
2. With a ruler and a dark marker, draw a line
hyperbola
across the midsection of the piece of paper.
circle
3. Mark a point anywhere on the paper, except
on the line. Label the point A.
ellipse
4. Mark another point anywhere on the line,
and label it B.
5. Using the ruler, draw line segment AB. Fold
the paper so that point A lies directly on top
of point B.
A
6. Crease the paper so that you can see the fold mark when the
paper has been flattened out. To see the fold mark better, use
the ruler and a pencil to draw a line along it. Make a
conjecture about the relationship between line segment AB
and the fold line.
B
7. Fold the paper so that point A falls on the original line but
not on point B. Make a crease so that you can easily see the
fold mark. Draw a line over the fold mark in pencil.
8. Repeat step 7 about 10 more times. Fold point A to a different point
on the original line each time. Be sure to choose an equal number
of points to the left and to the right of point B.
9. Describe the location of the parabola.
NEL
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Factored Form of a Quadratic
Relation
YOU WILL NEED
GOAL
• grid paper and ruler, or
Relate the factors of a quadratic relation to the key features
of its graph.
graphing calculator
INVESTIGATE the Math
Boris runs a dog kennel. He has
purchased 80 m of fencing to
build an outdoor exercise pen
against the wall of the kennel.
Career Connection
Jobs at a dog kennel include
kennel technician, veterinary
technician, consultant, groomer,
dog walker, and secretary.
exercise pen
kennel
?
What dimensions should Boris use to maximize the area
of the exercise pen?
A.
If x represents the width of the pen, write an expression for its length.
B.
Write a relation, in terms of x, for the area of the exercise pen. Identify
the factors of the relation.
C.
Create a table of values, and graph the relation you wrote for part B.
D.
Use your table of values or graph to verify that the area relation
is quadratic.
E.
Does the relation have a maximum value or a minimum value? Explain
how you know.
F.
Determine the zeros of the parabola.
G.
Determine the equation of the axis of symmetry of the parabola.
H.
Determine the vertex of the parabola.
I.
What are the dimensions that maximize the area of the exercise pen?
Reflecting
factored form of a quadratic
relation
J.
How are the factors of this relation related to the zeros of the graph?
K.
The area relation can also be written as A = - 2(x)(x - 40) or
A = - 2(x - 0)(x - 40), by dividing out the common factor of -2
from one of the factors. Explain why the factored form of a quadratic
relation is useful when graphing the relation by hand.
L.
What is the area of the largest exercise pen that Boris can build?
a quadratic relation that
is written in the form
y a(x r)(x s)
150
3.3 Factored Form of a Quadratic Relation
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3.3
APPLY the Math
EXAMPLE 1
Reasoning about the nature of a relation
Is the graph of y = 2(x + 1)(x - 5) a parabola? If so, in what direction
does it open? Justify your answer.
Jasper’s Solution
x
y
3
32
2
14
1
0
First
Difference
14 32 18
0 10
1 16
2 –18
14
Second
Difference
14 (18) 4
10
6
2
3 –16
4
4
I created a table of values. Then I calculated the
first and second differences. The second differences
are constant but not zero, and they are also
positive.
4
4
2
I predict that the graph of this relation is
a parabola that opens upward.
I used a graphing calculator to graph the relation
and check my predictions.
My predictions were correct.
EXAMPLE 2
Selecting a strategy to graph a quadratic relation given in factored form
Determine the y-intercept, zeros, axis of symmetry, and vertex of the
quadratic relation y = 2(x - 4)(x + 2). Then sketch the graph.
Cindy’s Solution
y = 2(x - 4)(x + 2)
y = 2(0 - 4)(0 + 2)
y = 2( - 4)(2)
y = - 16
The y-intercept occurs at (0, 16).
NEL
To determine the y-intercept, I substituted x 0
into the equation. I noticed that multiplying the
numbers in the original equation would have given
the same result.
Chapter 3
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0 = 2(x - 4)(x + 2)
x - 4 = 0 or x + 2 = 0
x = 4
x = -2
The zeros occur at (4, 0) and (2, 0).
4 + ( - 2)
2
x = 1
The equation of the axis of symmetry
is x = 1.
x =
y = 2(x - 4)(x + 2)
y = 2(1 - 4)(1 + 2)
y = 2( - 3)(3)
y = - 18
The vertex is (1, 18).
2
Page 152
To determine the zeros, I let y 0. I know that a
product is zero only when one of its factors is zero,
so I set each factor equal to 0 and solved for x.
The axis of symmetry passes through the midpoint
of the zeros, so I calculated the mean.
The vertex lies on the axis of symmetry, so its
x-coordinate is 1. I substituted x = 1 into
the equation of the parabola to determine
the y-coordinate.
y
x
-8 -6 -4 -2
0
-2
2
4
6
8
-4
I plotted the y-intercept, zeros, and vertex.
Then I joined the points with a smooth curve.
-6
-8
-10
-12
-14
-16
y 2(x 4)(x 2)
-18
EXAMPLE 3
Selecting a strategy to graph a quadratic relation given in factored form
Determine the y-intercept, zeros, axis of symmetry, and vertex of the
quadratic relation y = (x - 2)2. Then sketch the graph.
Kylie’s Solution
y = (x - 2)2
y = (0 - 2)2
y = 4
The y-intercept occurs at (0, 4).
152
3.3 Factored Form of a Quadratic Relation
To determine the y-intercept, I substituted x = 0
into the equation and solved for y.
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3.3
y = (x - 2)2
0 = (x - 2)2
0 = x - 2
x = 2
The zero occurs at (2, 0).
To determine the zeros, I let y = 0 and solved for x.
Both factors are the same, since y = (x - 2)2 is the
same as y = (x - 2)(x - 2). There is only one
solution to 0 = x - 2, so there is only one zero for
the quadratic relation.
The equation of the axis of symmetry is x = 2.
The axis of symmetry passes through the midpoint
of the zeros. Since there is only one zero, the axis
of symmetry must pass through it.
The vertex is (2, 0).
Since (2, 0) is on the line x = 2, this point is also
the vertex.
y
8
The y-intercept, (0, 4), is 2 units to the left of the
axis of symmetry. There must be another point with
y-coordinate 4 on the parabola, 2 units to the right
of x = 2. This point is (4, 4).
y = (x 2)2
6
(0, 4) 4
(4, 4)
2
(2, 0)
-4 -2
0
2
-2
4
x
6
I plotted these three points and joined them with a
smooth curve.
8
-4
x=2
EXAMPLE 4
Connecting the features of a parabola to its equation
Determine an equation for this parabola.
y
10
8
6
4
2
x
-4
-3 -2
-1
0
-2
1
2
3
4
Petra’s Solution
The zeros occur at (2, 0) and (1, 0).
y = a(x - r)(x - s)
y = a3x - ( - 2)4(x - 1)
y = a(x + 2)(x - 1)
NEL
I determined the zeros of the parabola and
substituted them into the factored form of a
quadratic relation. I did this because I know that
a parabola is described by a quadratic relation.
Chapter 3
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y
Page 154
There are infinitely many parabolas with these
zeros, all with different y-intercepts. A few
examples are shown in the diagram. To determine
the equation of the given parabola, I need to
determine the value of a. This is the only value that
varies in my equation y = a(x + 2)(x - 1).
10
8
6
4
2
x
-4
-3 -2
-1
0
-2
1
2
3
4
-4
y-intercept occurs at (0, 10).
y = a(x + 2)(x - 1)
10 = a(0 + 2)(0 - 1)
10 = a(2)(- 1)
10 = - 2a
-5 = a
I chose the y-intercept to substitute into my
equation because its coordinates are integers.
Then I solved for a.
An equation for the given parabola is
y = - 5(x + 2)(x - 1).
I substituted the value of a into my equation.
In Summary
Key Ideas
• When a quadratic relation is expressed in factored form
y = a(x - r)(x - s), each factor can be used to determine a zero, or
x-intercept, of the parabola.
• An equation for a parabola can be determined using the zeros and the
coordinates of one other point on the parabola.
Need to Know
• If a quadratic relation is
expressed in the form
y = a(x - r)(x - s),
• the x-intercepts are r and s
• the equation of the axis of
symmetry is the vertical
line defined by the
equation x = (r + s) , 2
• the x-coordinate of the
vertex is (r + s) , 2
• the y-intercept is
c = a * r * s
154
3.3 Factored Form of a Quadratic Relation
y a0
y a0
c
r
s
x
r
s
x
c
x (r s) 2
x (r s) 2
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3.3
CHECK Your Understanding
1. Complete the following for each quadratic relation below.
i) Determine the zeros.
ii) Explain how the zeros are related to the factors in the quadratic
expression.
iii) Determine the y-intercept.
iv) Determine the equation of the axis of symmetry.
v) Determine the coordinates of the vertex.
vi) Is the graph a parabola? How can you tell?
vii) Sketch the graph.
a) y = - 2x(x + 3)
b) y = (x - 3)(x + 1)
c) y = 2(x - 1)(x + 2)
2. Match each quadratic relation with the correct parabola.
a) y = (x - 2)(x + 3)
b) y = (x - 3)(x + 2)
c) y = (x + 2)(x + 3)
d) y = (3 - x)(x + 2)
e) y = (3 + x)(2 - x)
f ) y = (x - 2)(x - 3)
y
i)
y
iii)
8
8
6
6
4
4
2
2
v)
ii)
0
-2
2
x
-4 -2
4
x
2
-2
y
0
iv)
2
4
2
4
6
-4 -2
NEL
-2
2
4
-2
-4
-4
-6
-6
4
y
6
4
x
-4 -2
2
-2
vi)
2
0
0
-4
y
x
0
y
4
x
-8 -6 -4
6
2
2
4
x
-4 -2
0
-2
2
Chapter 3
4
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3. A quadratic relation has an equation of the form y = a(x - r)(x - s).
The graph of the relation has zeros at (2, 0) and (6, 0) and passes
through the point (3, 5). Determine the value of a.
PRACTISING
4. Determine the y-intercept, zeros, equation of the axis of symmetry, and
vertex of each quadratic relation.
a) y = (x - 3)(x + 3) d) y = - (x - 2)(x + 2)
b) y = (x + 2)(x + 2) e) y = 2(x + 3)2
c) y = (x - 2)(x - 2) f ) y = - 4(x - 4)2
5. Sketch the graph of each relation in question 4.
6. A quadratic relation has an equation of the form y = a(x - r)(x - s).
Determine the value of a when
a) the parabola has zeros at (4, 0) and (2, 0) and a y-intercept at (0, 1)
b) the parabola has x-intercepts at (4, 0) and ( - 2, 0) and
a y-intercept at (0, -1)
c) the parabola has zeros at (5, 0) and (0, 0) and a minimum value of - 10
d) the parabola has x-intercepts at (5, 0) and ( -3, 0) and a maximum
value of 6
e) the parabola has its vertex at (5, 0) and a y-intercept at (0, - 10)
7. Determine the zeros, equation of the axis of symmetry, and vertex of
K
each parabola. Then determine an equation for each quadratic relation.
a)
c)
y
y
80
4
40
2
x
0
-80 -40
-40
x
40 80
-4 -2
-80
-2
4
20
2
10
0
x
x
-10
-10
3.3 Factored Form of a Quadratic Relation
4
y
d)
30
0
156
2
-4
y
b)
0
10 20 30
-4 -2
-2
2
-4
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3.3
8. a) Sketch the graph of y = a(x - 2)(x + 3) when a = 3.
b) Describe how your graph for part a) would change if the value of a
changed to 2, 1, 0, - 1, -2, and -3.
9. a) Sketch the graph of y = (x - 2)(x - s) when s = 3.
b) Describe how your graph for part a) would change if the value of s
changed to 2, 1, 0, 1, -2, and -3.
10. The x-intercepts of a parabola are -3 and 5. The parabola crosses
the y-axis at - 75.
a) Determine an equation for the parabola.
b) Determine the coordinates of the vertex.
11. Sometimes the equation y = a(x - r)(x - s) cannot be used
to determine the equation of a parabola from its graph. Explain
when this is not possible, and draw graphs to illustrate.
12. A ball is thrown into the air from the roof of a building that is 25 m
high. The ball reaches a maximum height of 45 m above the ground
after 2 s and hits the ground 5 s after being thrown.
a) Use the fact that the relation between time and the height
of the ball is a quadratic relation to sketch an accurate graph
of the relation.
b) Carefully fold the graph along its axis of symmetry. Extend
the short side of the parabola to match the long side.
c) Where does the extended graph cross the time axis?
d) What are the zeros of the relation?
e) Determine the coordinates of the vertex.
f ) Determine an equation for the relation.
g) What is the meaning of each zero?
13. A car manufacturer decides to change the price of its new luxury sedan
(LS) model to increase sales. The graph shows the relationship between
revenue and the size of the price change.
y
Potential revenue
($10 000 000)
10
8
6
4
2
x
-200
-100
0
100
Amount of price change ($)
a) Determine an equation for the graph.
b) How should the price be changed for maximum revenue?
NEL
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14. Ryan owns a small music store. He currently charges $10 for each CD.
T
At this price, he sells about 80 CDs a week. Experience has taught him
that a $1 increase in the price of a CD means a drop of about five CDs
per week in sales. At what price should Ryan sell his CDs to maximize
his revenue?
15. Rahj owns a hardware store. For every increase of 10¢ in the price
A
Career Connection
Jobs in the music industry
include recording artist, studio
musician, songwriter, producer,
recording engineer, digital audio
workstation operator, music
programmer, and re-mixer.
of a package of batteries, he estimates that sales decrease by
10 packages per day. The store normally sells 700 packages of batteries
per day, at $5.00 per package.
a) Determine an equation for the revenue, y, when x packages of
batteries are sold.
b) What is the maximum daily revenue that Rahj can expect from
battery sales?
c) How many packages of batteries are sold when the revenue is at
a maximum?
16. Create a flow chart that summarizes the process you would use
C
to determine an equation of a parabola from its graph. Assume that
the parabola has two zeros.
Extending
17. Without graphing, match each quadratic relation in factored form
(column 1) with the equivalent quadratic relation in standard form
(column 2). Explain your reasoning.
Column 1
a) y = (2x - 3)(x + 4)
b) y = (3x + 1)(4x - 3)
c) y = (3 - 2x)(4 + x)
d) y = (3 - 4x)(1 + 3x)
Column 2
i)
ii)
iii)
iv)
v)
vi)
y = 12x 2 - 5x - 3
y = - 2x 2 - 5x + 12
y = 2x 2 + 11x - 12
y = 2x 2 + 5x - 12
y = - 12x 2 + 5x + 3
y = 12x 2 + 5x - 3
18. Martin wants to enclose the backyard of his house on three sides to
form a rectangular play area. He is going to use the wall of his house
and three sections of fencing. The fencing costs $15/m, and Martin
has budgeted $720. Determine the dimensions that will produce
the largest rectangular area.
158
3.3 Factored Form of a Quadratic Relation
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
What are the key properties of a quadratic relation?
A:
The key properties are:
• In a table of values, the second differences are constant and not zero.
• The degree of the equation that represents the relation is 2.
• The graph has a U shape, which is called a parabola.
• Every parabola has a vertex that is the highest or lowest point
on the curve.
• Every parabola has an axis of symmetry that passes through its vertex.
Q:
A:
What information can you easily determine from the
factored and standard forms of a quadratic relation?
From the standard form y = ax 2 + bx + c, you can determine
the y-intercept, which is c.
From the factored form y = a(x - r)(x - s), you can determine
• the zeros, or x-intercepts, which are r and s
r + s
• the equation of the axis of symmetry, which is x =
2
• the coordinates of the vertex, by substituting the value of the axis
of symmetry for x in the relation
• the y-intercept, which is a * r * s
Study
Aid
• See Lesson 3.1 and
Lesson 3.2, Examples 1 to 4.
• Try Mid-Chapter Review
Questions 1 and 2.
Study
Aid
• See Lesson 3.2, Example 2,
and Lesson 3.3,
Examples 1 to 3.
• Try Mid-Chapter Review
Questions 3 to 7.
From both forms, you can determine the direction in which the
parabola opens: upward when a 7 0 and downward when a 6 0.
Q:
If you are given information about a quadratic relation,
how can you determine the equation?
A:
If the graph has zeros, these can be used to write the equation of the
quadratic relation in factored form. Then you can use a different
point on the parabola to determine the coefficient a.
Study
Aid
• See Lesson 3.3, Example 4.
• Try Mid-Chapter Review
Questions 8 to 10.
EXAMPLE
The points (⫺2, 0) and (3, 0) are the zeros of a parabola that passes
through (4, 12). Determine an equation for the quadratic relation.
Solution
Use the zeros to write the equation y = a(x + 2)(x - 3). Substitute the
coordinates of the point (4, 12) into the equation to determine the coefficient a.
12 = a(4 + 2)(4 - 3)
12 = a(6)(1)
2 = a
An equation for the quadratic relation is y = 2(x + 2)(x - 3).
NEL
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PRACTICE Questions
Lesson 3.1
6. A soccer ball is kicked into the air. Its height,
1. State whether each relation is quadratic. Justify
your answer.
a)
y
2
1
x
-2
0
-1
1
-1
2
-2
b) y = 5x 2 + 3x - 1
c)
x
0
1
2
y
3
2
1
3
4
5
0
1
2
2. Each table of values represents a quadratic
relation. Decide, without graphing, whether the
parabola opens upward or downward.
a)
x ⫺2 ⫺1
0
1
2
b)
y
0
⫺5
0
15
40
x
⫺2
⫺1
0
1
2
y
⫺3
3
5
3
⫺3
Lesson 3.2
3. Graph y = - x 2 + 6x to determine
a) the equation of the axis of symmetry
b) the coordinates of the vertex
c) the y-intercept
d) the zeros
h, in metres, is approximated by the equation
h = - 5t 2 + 15t + 0.5, where t is the time
in seconds since the ball was kicked.
a) From what height is the ball kicked?
b) When does the ball hit the ground?
c) When does the ball reach its maximum
height?
d) What is the maximum height of the ball?
e) What is the height of the ball at t = 3?
Is the ball travelling upward or downward
at this time? Explain.
f ) When is the ball at a height of 10 m?
Lesson 3.3
7. Determine the y-intercept, zeros, equation of the
axis of symmetry, and vertex of each quadratic
relation. Then sketch its graph.
a) y = (x - 5)(x + 5)
b) y = - (x - 6)(x - 2)
c) y = 2(x - 1)(x + 3)
d) y = - 0.5(x + 4)2
8. The zeros of a parabola are ⫺10 and 30.
The parabola crosses the y-axis at 50.
a) Determine an equation for the parabola.
b) Determine the coordinates of the vertex.
9. Determine an equation for this quadratic
relation.
8
y
6
4
4. The points (⫺3, 8) and (9, 8) lie on opposite
2
sides of a parabola. Determine the equation
of the axis of symmetry.
0
x
-8 -6 -4 -2
-2
2
5. Use a graphing calculator to graph each relation.
Determine the y-intercept, zeros, equation of the
axis of symmetry, and vertex.
a) y = x 2 + 8x + 15
b) y = - 2x 2 + 16x - 32
160
Mid-Chapter Review
10. Give an example of an equation of a quadratic
relation whose vertex and x-intercept occur at
the same point.
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Expanding Quadratic
Expressions
YOU WILL NEED
GOAL
Determine the product of two binomials using a variety of strategies.
• algebra tiles
LEARN ABOUT the Math
y
10
Brandon was doing his math homework. For one question, he had
to determine the equation of the parabola shown at the right.
8
6
Brandon’s answer was y = (x + 4)(x + 2).
4
His older sister, Devin, said that the answer can also be y = x2 + 6x + 8.
?
2
x
How can Devin show Brandon that both answers are correct?
-8 -6 -4 -2
EXAMPLE 1
Connecting an area model to the product
of two binomials
0
-2
2
4
Show that the equations y = (x + 4)(x + 2) and y = x2 + 6x + 8
represent the same quadratic relation.
Devin’s Solution
y = (x + 4)(x + 2)
x
1
1
1
1
x
1
1
1
1
x
x2
x
x
x
x
1
x
1
1
1
1
1
x
1
1
1
1
x
I wanted to show Brandon
how to multiply two binomials.
I know that the area of a
rectangle is the product of
its length and its width. I used
algebra tiles to represent a width
of x 2 and a length of x 4.
1
Communication
Tip
The tiles used for each
dimension of a rectangle can
be placed on either the left
or right, and either on the top
or bottom. The resulting area
of the rectangle is the same in
each case. The only difference
occurs in the position of the
x 2, x, and unit tiles within
the rectangle. For example,
1
NEL
I used x2 tiles, x tiles, and unit
tiles to fill in the area of the
rectangle with these dimensions.
The area of the rectangle
represents the product of
the two binomials.
and
represent the same product.
Chapter 3
161
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x2
x
x
x
x
x
1
1
1
1
x
1
1
1
1
I counted the tiles in the
rectangle to get an expression
for its area, A.
A = x 2 + 2x + 4x + 8
A = x 2 + 6x + 8
y = (x + 4)(x + 2) and
y = x 2 + 6x + 8 are the same
quadratic relation.
Brandon’s equation is in factored
form and mine is in standard
form.
I graphed both relations
to see if they represented the
same parabola. The second
parabola traced exactly over
the first parabola.
Reflecting
x
4
A.
Why did Devin use only red tiles in her rectangle model?
x
x2
4x
B.
Explain how the area diagram at the left is related to Devin’s algebra
tile model and the product (x + 4)(x + 2).
2
2x
8
C.
Is the value of a always the same in factored form and standard form
if both relations represent the same parabola? Explain.
APPLY the Math
EXAMPLE 2
Connecting the product of two binomials to the distributive property
Expand and simplify.
a) (2x + 3)(x - 2)
b) (2x - 1)(x - 3)
Lorna’s Solution
a)
x
x
x
1
1
1
x2
x2
x
x
x
1
x
x
1 1 1
1
x
x
1 1 1
162
3.4 Expanding Quadratic Expressions
I placed tiles that correspond to the binomial factors along
the sides of a rectangle. I represented x 2 as x (2)
because I didn’t know how to remove part of a tile.
Then I used tiles to fill in the area. The rules for multiplying
integers helped me choose the correct colours to use.
Since a blue tile is negative and a red tile is positive, I used
blue tiles to represent the negative product.
NEL
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3.4
(2x + 3)(x - 2) = 2x 2 - 4x + 3x - 6
= 2x 2 - x - 6
2x
3
I noticed that the area in the tile model was divided
into four sections, so I divided a rectangle into four
small rectangles. I labelled the side lengths.
x
2
2x
3
x
2x2
3x
2
4x
6
(2x + 3)(x - 2) = 2x 2 - 4x + 3x - 6
= 2x 2 - x - 6
(2x + 3)(x - 2) = 2x(x - 2) + 3(x - 2)
= 2x 2 - 4x + 3x - 6
= 2x 2 - x - 6
b)
x
x
x2
x2
x
1
x
x
1
1
x
x
1
1
x
x
1
2x
1
x
2x2
x
3
6x
3
I wrote an expression for the area of each small
rectangle. The area of the large rectangle is the
sum of the areas of the four small rectangles.
When I collected like terms, I saw that the product
was the same.
I recognized the distributive property in the area
model. The areas in the first column show the
product 2x(x 2). The areas in the second column
show the product 3(x 2). I used the distributive
property again. Then I collected like terms to get
the final result.
1
x
(2x - 1)(x - 3) = 2x(x - 3) -1(x - 3)
= 2x 2 - 6x - x + 3
= 2x 2 - 7x + 3
NEL
I counted the different types of algebra tiles to get
the product.
I created an algebra tile model. This time the unit
tiles that I used to fill in the area had to be positive
red tiles, since the result is the product of two
negative blue tiles.
I made an area diagram to show the area
of the four sections of the tile model.
I could have used the distributive property without
a picture or model. I collected like terms to get the
final result.
Chapter 3
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Representing the product of two binomials symbolically
Multiply each expression.
a) (x - 5)(x + 5)
b) (3x - 5)2
Zac’s Solution
a) (x - 5)(x + 5) = x 2 + 5x - 5x - 25
= x 2 - 25
I multiplied each term in the second binomial by x
and then by 5. I collected like terms and got a
binomial for my final result.
b) (3x - 5)2 = (3x - 5)(3x - 5)
I wrote the expression as a product of two
binomials. I multiplied each term in the second
binomial by 3x and then by 5. I collected like
terms and got a trinomial for my final result.
= 9x 2 - 15x - 15x + 25
= 9x 2 - 30x + 25
EXAMPLE 4
Connecting the factored form and standard form of a quadratic relation
Determine the equation of the parabola.
Express your answer in standard form.
y
4
2
x
-6 -4 -2
0
-2
2
4
-4
-6
Mathieu’s Solution
y = a[x - ( - 4)](x - 2)
y = a(x + 4)(x - 2)
I wrote the equation in factored form using the
zeros of the parabola. Then I wrote an equivalent
expression for x (4).
x = 0, y = 4
4 = a(0 + 4)(0 - 2)
4 = a(4)(- 2)
4 = - 8a
4
-8a
=
-8
-8
- 0.5 = a
There is only one value of a that gives a parabola
with these zeros and y-intercept. To determine this
value, I substituted the coordinates of the y-intercept
(0, 4) into the equation and solved for a.
164
3.4 Expanding Quadratic Expressions
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3.4
y = - 0.5(x + 4)(x - 2)
y = - 0.5(x 2 - 2x + 4x - 8)
y = - 0.5x 2 + x - 2x + 4
y = - 0.5x 2 - x + 4
I substituted the value of a into the factored form of
the equation. I multiplied the two binomials. Then I
multiplied all the terms by 0.5 and collected like
terms to get the result in standard form.
In Summary
Key Ideas
• Quadratic expressions can be expanded using the distributive property,
then simplified by collecting like terms.
• An area diagram or algebra tiles can be used to show the relation
between two binomial factors of degree one and their product.
Need to Know
• To calculate the product of two binomials, use the distributive property
twice.
ax
b
cx
acx2
bcx
d
adx
bd
(ax b)(cx d ) ax(cx d ) b(cx d )
acx2 adx bcx bd
CHECK Your Understanding
1. State the binomials that are represented by the length and width of each
rectangle. Then determine the product that is represented by the area.
a)
NEL
x
1
1
1
1
1
x
x2
x
x
x
x
x
1
x
1
1
1
1
1
b)
x
1 1
x
x2
x x
1
x
1
1
1
x
1
1
Chapter 3
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2. Copy and complete this table.
Expanded and
Simplified Form
Expression
Area Diagram
(x 2)(x 3)
x
3
x
x2
3x
2
2x
6
x2 5x 6
a) (x 1)(x 6)
b) (x 1)(x 4)
c) (x 2)(x 2)
d) (x 3)(x 4)
e) (x 2)(x 4)
f ) (x 2)(x 6)
PRACTISING
3. Determine the missing terms.
a) (m + 3)(m + 2) = . + 2m + 3m + 䊉
b) (k - 2)(k + 1) = . + 䊉 - 2k - 2
c) (r + 4)(r - 3) = r 2 - 3r + . - 䊉
d) (x - 5)(x - 2) = x 2 - . - 䊉 + 10
e) (2n + 1)(3n - 2) = . - 䊉 + 3n - 2
f ) (5m - 2)(m - 3) = 5m 2 - . - 2m + 䊉
4. Expand and simplify.
a) (x + 2)(x + 5)
b) (x + 2)(x + 1)
c) (x + 2)(x - 3)
d) (x + 2)(x - 1)
e) (x - 4)(x - 2)
f ) (x - 5)(x - 3)
a) (5x + 2)(x + 2) c) (x - 2)(7x + 3)
b) (x + 2)(4x + 1) d) (3x - 2)(x + 1)
e) (x - 2)(4x - 6)
f ) (7x - 5)(x - 3)
5. Expand and simplify.
6. Expand and simplify.
a) (x + 3)(x - 3)
b) (x + 6)(x - 6)
c) (2x - 1)(2x + 1) e) (4x - 6)(4x + 6)
d) (3x - 3)(3x + 3) f ) (7x - 5)(7x + 5)
7. Expand and simplify.
a) (x + 1)2
b) (a + 4)2
166
3.4 Expanding Quadratic Expressions
c) (c - 1)2
d) (5y - 2)2
e) (6z - 5)2
f ) (–3d + 5)2
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3.4
8. Write a simplified expression for the area of each figure.
A
a)
c)
5x 3
4m 4
2x 4
2m 3
b)
x1
d)
3m 2
2x 2
3x 1
9. Expand and simplify.
a) 4(x - 6)(x + 7)
b) - (x + 3)(4x - 1)
c) 6x(x + 1)2
d) (x + 4)(x - 2) + (x - 1)(x + 5)
e) (4x - 1)(4x + 1) - (x + 3)2
f ) 2(3x + 4)2 - 3( x - 2)2
10. Expand and simplify.
a) (x + y)(2x + 3y)
b) (x + 2y)(3x + y)
c) (3x - 2y)(5x + 4y)
d) (8x - y)(7x + 2y)
e) (6x - 5y)(6x + 5y)
f ) (9x - 7y)2
11. Determine the equation of each parabola. Express the equation
K
in standard form.
a)
4
c)
y
y
4
2
2
x
-6 -4 -2
0
2
-2
4
x
6
-4 -2
-4
2
-2
4
6
8
-4
-6
-6
-8
b)
0
-8
4
d)
y
4
2
y
2
x
-8 -6 -4 -2
NEL
0
-2
2
4
x
-4 -2
0
-2
-4
-4
-6
-6
-8
-8
2
4
6
8
Chapter 3
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12. Write each quadratic relation in standard form. State which way
the parabola opens.
Zeros
A Point on the Graph
a)
1 and 7
(3, 5)
b)
1 and 5
(3, 4)
c)
3 and 7
(0, 3)
d)
2 and 6
(1, 1)
e)
2 and 8
(3, 7)
13. The area of a rectangle is represented by the expression
2x 2 + 14x + 20. Bill claims that this rectangle could have either
the dimensions (2x + 4) and (x + 5) or the dimensions (2x + 10)
and (x + 2). Do you agree or disagree? Justify your opinion.
14. Explain how you know that the product will be quadratic when you
C
expand (12x - 7)(5x + 1).
15. The Rainbow Bridge in Utah, shown at the left, is a natural arch that
T
is approximately parabolic in shape. The arch is about 88 m high. It is
84 m across at its base. Determine a quadratic relation, in standard
form, that models the shape of the arch.
16. Jay claims that whenever two binomials are multiplied together,
the result is always a trinomial. Is his claim correct? Use examples
to support your decision.
Extending
17. Expand and simplify each expression.
a) (x + 3)3
b) (2x - 2)3
c) (4x + 2y)3
d) [(x + 2)(x - 2)]2
e) (x + 6)(x + 3)(x - 6)(x - 3)
f ) (3x 2 + 6x - 1)2
18. Expand each expression.
a) (a + b)1
b) (a + b)2
c) (a + b)3
d) (a + b)4
19. Discuss any patterns you see in question 18.
168
3.4 Expanding Quadratic Expressions
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Quadratic Models Using
Factored Form
YOU WILL NEED
• graphing calculator
• grid paper
• ruler
GOAL
Determine the equation of a quadratic model using the factored
form of a quadratic relation.
3 points
INVESTIGATE the Math
You can draw one straight line through any pair of points. If you have three
points you can draw a maximum of three lines. The maximum number of
lines possible occurs when the points do not lie on the same line.
?
What is the maximum number of lines you can draw using
100 points?
A.
Can you answer the question directly using a diagram? Explain.
B.
Since two points are needed to draw a line, using zero and one point
results in zero lines. Copy and complete the rest of the table by
drawing each number of points and determining the maximum
number of lines that can be drawn through pairs of points.
Number of Points, x
0
1
Maximum Number of Lines, y
0
0
2
3
4
5
6
C.
Use your data to create a scatter plot with an appropriate scale.
D.
What shape best describes your graph? Draw a curve of good fit.
E.
Carry out appropriate calculations to determine whether the curve you
drew for part D is approximately linear, approximately quadratic, or
some other type.
F.
What are the zeros of your curve? Use the zeros to write an equation
for the relation in factored form: y = a(x - r)(x - s).
G.
Use one of the ordered pairs in your table (excluding the zeros) to
calculate the value of a. Write an equation for the relation in both
factored form and standard form.
quadratic regression
Use a graphing calculator and quadratic regression to determine the
equation of this quadratic relation model.
a process that fits the
second degree relation
y ax2 bx c to the data
H.
NEL
curve of good fit
a curve that approximates, or is
close to, the distribution of
points in a scatter plot
Chapter 3
169
curve of best fit
I.
the curve that best describes the
distribution of points in a scatter
plot, usually found using a
process called regression
How does your equation compare with the graphing calculator’s
curve of best fit equation?
J.
Use your equation to predict the number of lines that can be drawn
using 100 points.
Tech
Support
For help using a TI-83/84
graphing calculator to
determine the equation of a
curve of best fit using quadratic
regression, see Appendix B-10.
If you are using a TI-nspire,
see Appendix B-46.
Reflecting
K.
How does the factored form of a quadratic relation help you determine
the equation of a curve of good fit when it has two zeros?
L.
How would the equation change if the data were quadratic and
the curve of good fit had only one zero?
M. If a curve of good fit for a set of data had no zeros, could the factored
form be used to determine its equation? Explain.
APPLY the Math
EXAMPLE 1
Connecting the zeros and factored form
to an equation that models data
Data from the flight of a golf ball are given in this table. If the maximum
height of the ball is 30.0 m, determine an equation for a curve of good fit.
Horizontal Distance (m)
0
30
60
80
90
Height (m)
0.0
22.0
30.0
27.0
22.5
Jill’s Solution
36
y
I plotted the data. I drew a
parabola as a curve of good fit
because it seemed to be close to
most of the data. The maximum
height was 30.0 m, so the vertex
of the parabola is located at
(60, 30). The equation of the axis
of symmetry is x ⫽ 60.
32
28
Height (m)
24
20
16
12
8
4
0
170
x
20 40 60 80 100 120 140
Horizontal distance (m)
3.5 Quadratic Models Using Factored Form
A zero occurs at (0, 0). Since
a parabola is symmetric, I
determined that another zero
is located at (120, 0).
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3.5
y = a(x - 0)(x - 120)
30 = a(60)(60 - 120)
30
= a
60( - 60)
1
= a
2( - 60)
1
= a
120
1
y = x(x - 120)
120
1 2
y = x + x
120
When x = 30,
1
y = -a
b (30)(30 - 120)
120
I wrote a general equation of the
parabola in factored form.
Because the zeros are 0 and
120, I knew that (x 0) and
(x 120) are factors.
I substituted (60, 30) into the
equation, since it is a point on
the curve. Then I solved for a.
I used the value of a to write
the equation. Then I expanded
the equation to write it in
standard form.
I checked the equation by
substituting other values of x
into it.
1
y = - a b (- 90)
4
1
y = 22
2
When x = 80,
1
y = -a
b (80)(80 - 120)
120
2
y = - a b (- 40)
3
2
y = 26
3
1
The points a 30, 22 b and
2
2
a80, 26 b from the equation
3
are close to the points (30, 22.0)
and (80, 27.0) from the data.
An equation of good fit is y = NEL
The results for y were close to
the values in the table, so the
equation for the curve of good
fit is reasonable.
1 2
x + x.
120
Chapter 3
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EXAMPLE 2
Selecting an informal strategy to determine
an equation of a curve of good fit
A competitive diver does a handstand dive from a 10 m platform. This
table of values shows the time in seconds and the height of the diver,
relative to the surface of the water, in metres.
Time (s)
0
0.3
0.6
0.9
1.2
1.5
Height (m)
10.00
9.56
8.24
6.03
2.94
1.03
Determine an equation that models the height of the diver above
the surface of the water during the dive. Verify your result using
quadratic regression.
Madison’s Solution
Tech
I entered the data in the lists of
a graphing calculator and
created a scatter plot.
Support
For help using a TI-83/84
graphing calculator to create
a scatter plot, see Appendix
B-10. If you are using a
TI-nspire, see Appendix B-46.
The points looked like they
formed half of a parabola. I
assumed that the diver was at
the maximum height at the start
of the dive. This meant the
vertex was located at (0, 10).
I estimated that one zero
occurred at 1.4 and the other
zero occurred at 1.4 since the
y-axis is the axis of symmetry.
y = a(x - 1.4)(x + 1.4)
y = - 1(x - 1.4)(x + 1.4)
I wrote an equation of the
parabola in factored form.
I entered my equation into the
equation editor using 1 as a
guess for the value of a. I knew
that a is negative since the
parabola opens downward.
This graph is not a good fit.
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3.5 Quadratic Models Using Factored Form
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3.5
y = - 3(x - 1.4)(x + 1.4)
I tried 3 as a value of a and
graphed the relation again.
This graph is not a good fit either.
y = - 5(x - 1.4)(x + 1.4)
I tried 5 as a value of a and
graphed the relation again.
This graph is a good fit that models
the height of the diver above the
surface of the water during the dive.
y = - 5(x 2 + 1.4x - 1.4x - 1.96)
y = - 5x 2 + 9.8
I expanded my equation to write
it in standard form.
Then I used quadratic regression
to determine the equation of
the curve of best fit. My
equation and the calculator's
equation are very close.
EXAMPLE 3
Solving a problem using a model
Jeff and Tim are analyzing data collected from a motion detector following
the launch of their model rocket.
Time (s)
0.0
1.0
2.0
3.0
4.0
Height (m)
0.0
16.0
20.0
15.5
0.0
a) Determine an equation for a curve of good fit.
b) Use the equation you determined for part a) to estimate the height
of the rocket 0.5 s after it is launched.
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Phil’s Solution
Since the height of the rocket increased and then
decreased, I assumed that a quadratic model might
be reasonable.
a) A parabola might model
this situation.
y = a(x - 0)(x - 4)
y = ax(x - 4)
20 = a(2)(2 - 4)
To determine the value of a, I substituted (2, 20)
into the equation since it is a point on the curve.
20
= a
(2)( - 2)
-5 = a
I used the value of a to write the final equation
of the curve of good fit.
y = - 5x(x - 4)
When x = 2,
y = - 5(2)(2 - 4)
y = - 5(2)( - 2)
y = 20
I wrote a general equation of the relation in
factored form. The zeros are 0 and 4, so (x 0)
and (x 4) are factors.
When x = 3,
y = - 5(3)(3 - 4)
y = - 5(3)( -1)
y = 15
I checked my equation by substituting other
values of x into it. The results for y were close
to the values in the table so my equation
seems reasonable.
I substituted 0.5 for x into the equation for the
curve of good fit.
b) When x = 0.5,
y = - 5(0.5)(0.5 - 4)
y = - 5(0.5)( - 3.5)
y = 8.75
The height of the rocket after 0.5 s is
approximately 8.8 m.
y
I checked the result by plotting the points and
drawing the graph. The fit seems reasonable.
Height (m)
20
16
12
8
4
0
174
x
1
2 3
Time (s)
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3.5 Quadratic Models Using Factored Form
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3.5
In Summary
Key Idea
• If a curve of good fit for data with a parabolic pattern passes through
the horizontal axis, then the factored form of the quadratic relation can
be used to determine an algebraic model for the relationship.
Need to Know
• The estimated or actual x-intercepts, or zeros, of a curve of good fit
represent the values of r and s in the factored form of the quadratic
relation y a(x r)(x s).
• The value of a can be determined algebraically by substituting the
coordinates of a point (other than a zero) that lies on or close to the
curve of good fit into the equation and then solving for a.
• The value of a can be determined graphically by estimating the value
of a and graphing the resulting parabola with graphing technology. By
observing the graph, you can adjust your estimate of a and graph again
until the parabola passes through or close to a large number of points
in the scatter plot.
• Graphing technology can be used to determine an algebraic model for
the curve of best fit. You can use quadratic regression when the data
has a parabolic pattern.
CHECK Your Understanding
1. a) Use the graph at the right to determine an equation for a curve
y
of good fit. Write the equation in factored and standard forms.
b) Use your equation to estimate the value of y when x 1.
4
2
2. a) These data represent the path of a soccer ball as it flies
through the air. Create a scatter plot, and then determine
an equation for a quadratic curve of good fit.
-2
0
-2
2
4
6
8
x
10
-4
Horizontal Distance (m)
0.0
1.0
2.0
3.0
4.0
Height (m)
1.0
1.6
1.9
1.6
1.0
-6
-8
b) Use your equation for part a) to estimate the height of the ball
when its horizontal distance is 1.5 m.
3. a) Determine the equation of the quadratic curve of best fit for the data.
x
1
0
1
2
3
y
2.4
3.6
3.6
2.3
0.1
b) Use your equation for part a) to estimate the value of y
when x = 3.2 .
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PRACTISING
4. A parabola passes through the points (4, 10), (3, 0), (2, 6),
K (1, 8), (0, 6), (1, 0), and (2, 10).
a) Determine an equation for the parabola in factored form.
b) Express your equation in standard form.
c) Use a graphing calculator and quadratic regression to verify
the accuracy of the equation you determined.
5. A water balloon was launched from a catapult. The table shows the
data collected during the flight of the balloon using stop-motion
photography.
Horizontal
Distance (m) 0
Height (m)
6
12
18
24
30
36
42
48
54
0.0 11.6 20.4 26.4 29.5 29.7 27.1 21.6 13.3
2.1
a) Use the data to create a scatter plot. Then draw a curve of good fit.
b) Determine an equation for the curve you drew.
c) Estimate the horizontal distance of the balloon when it reached
its maximum height. Then use your equation to calculate
its maximum height.
d) Use your equation to determine the height of the balloon when
its horizontal distance was 40 m.
6. An emergency flare was shot into the air from the top of a building.
A
The table gives the height of the flare at different times during
its flight.
Time (s)
0
1
2
3
4
5
6
Height (m)
60
75
80
75
60
35
0
a) How tall is the building?
b) Use the data in the table to create a scatter plot. Then draw a curve
of good fit.
c) Determine an equation for the curve you drew.
d) Use your equation to determine the height of the flare at 2.5 s.
7. A hang-glider was launched from a platform on the top of the Niagara
Escarpment. The data describe the first 13 s of the flight. The values
for height are negative whenever the hang-glider was below the top
of the escarpment.
Time (s)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Height (m) 10.0 0.8 9.2 15.2 18.8 20.0 18.8 15.2 9.2 0.8 10.0 23.2 38.8 56.8
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3.5 Quadratic Models Using Factored Form
NEL
3.5
a) Determine the height of the platform.
b) Determine an equation that models the height of the hang-glider
over the 13 s period.
c) Determine the lowest height of the hang-glider and when
it occurred.
8. The data in the table at the right represent the height of a golf ball
at different times.
a) Create a scatter plot, and draw a curve of good fit.
b) Use your graph for part a) to approximate the zeros of the relation.
c) Determine an equation that models this situation.
d) Use your equation for part c) to estimate the maximum height
of the ball.
9. For a school experiment, Nichola recorded the height of a model
rocket during its flight. The motion detector stopped working,
however, during her experiment. The following data were collected
before the malfunction.
Time (s)
Height (m)
0.0
0.000
0.5
10.175
1.0
17.900
1.5
23.175
2.0
26.000
2.5
26.375
3.0
24.300
Time (s)
0.0
1.0
2.0
3.0
4.0
3.5
19.775
Height (m)
2.00
19.5
27.0
24.5
12.0
4.0
12.800
4.5
3.375
a) The height–time relation is quadratic. Determine an equation
for the height–time relation.
b) Use the equation you determined for part a) to estimate the height
of the rocket at 3.8 s.
c) Determine the maximum height of the rocket. When did
the rocket reach its maximum height?
10. A pendulum swings back and forth. The time taken to complete one
back-and-forth swing is called the period.
Period (s)
0.5
1.0
1.5
2.0
2.5
Length of
Pendulum (cm)
6.2
24.8
55.8
99.2
155.0
a) Can the data be represented by a quadratic relation? How do you
know?
b) Use the data to draw a scatter plot. Then sketch a curve of
good fit.
c) Assuming that your graph is a parabola with vertex (0, 0),
determine an equation for your curve of good fit.
d) Estimate the period for a pendulum that is 80.0 cm long.
e) Estimate the length of a pendulum that has a period of 2.3 s.
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11. Examine this square dot pattern.
T
Diagram 1
Diagram 2
Diagram 3
How many dots are in the 20th diagram? Justify your answer.
12. Examine these three figures made of squares.
Figure 1
Figure 2
Figure 3
a) Create a table of values to compare the figure number, x, with
the area, y. Draw figures 4 and 5 and add this data to your table.
b) Create a difference table to show that the relationship between
the figure number and the area is quadratic.
c) Determine an equation for this relationship.
d) Using your difference table, work backwards to determine
the zeros of this relationship.
e) Verify that the zeros you determined correspond to your equation.
f ) What restriction must be placed on x to model this relationship
accurately?
13. Can the factored form of a quadratic relation always be used to model
C
a curve of good fit for data that appear to be quadratic? Explain.
14. Create a flow chart that summarizes the steps for determining
the equation of a parabola of good fit using the factored form
of a quadratic relation.
Extending
15. Examine this pattern of cube structures.
Model 1
Model 2
Model 3
a) Determine the number of cubes in the 15th model.
b) Which model in this pattern could you build using 1249 cubes?
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3.5 Quadratic Models Using Factored Form
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Exploring Quadratic and
Exponential Graphs
YOU WILL NEED
• graphing calculator
GOAL
Compare the graphs of y x2 and y 2x to determine
the meanings of zero and negative exponents.
EXPLORE the Math
When you fold a piece of paper in half, you create two regions of equal
area. The number of regions increases each time you make a new fold.
?
What is the relation between the number of regions and the
number of folds, and how does it compare to y x2?
A.
Copy and complete the table. Discuss any patterns you see.
Number of Folds
1
Number of Regions
2
2
3
4
5
6
7
B.
Is the relationship between the number of regions and the number
of folds quadratic? Explain.
C.
If x represents the number of folds and y represents the number
of regions, show that the equation y = 2x fits the data you found.
D.
On a graphing calculator, enter the equation y = x 2 into Y1 of the
equation editor. Then enter y = 2x into Y2. Change the line to a
thick line for Y2. Use the window settings shown to graph both
relations.
Tech
E.
NEL
Discuss how the graphs are the same and how they are different.
Support
For help graphing relations and
changing window settings
using a TI-83/84 graphing
calculator, see Appendix B-2
and B-4. If you are using a
TI-nspire, see Appendix B-38
and B-40.
Chapter 3
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Tech
Support
For help with the Table feature
using a TI-83/84 graphing
calculator, see Appendix B-6.
If you are using a TI-nspire,
see Appendix B-42.
Tech
Support
To change a decimal to a
fraction on a TI-83/84
graphing calculator, return
to the home screen by pressing
2ND
MODE
5/12/09
ENTER
ENTER
.
Page 180
F.
Create a table of values using the Table feature. Use a starting value of
5 and an increment of 1. Scroll down the X column in your table to
compare the y-values of the two relations. Which relation grows faster
as x becomes greater?
G.
Scroll down the X column in your table. Find the corresponding
number in Y2 to determine the value of the power.
i) 0, to determine the value of 20
ii) 1, to determine the value of 21
iii) 2, to determine the value of 22
iv) 3, to determine the value of 23
H.
Express each decimal for part G as a fraction. Rewrite each fraction by
changing the denominator to a power of 2.
I.
Based on your answers for parts G and H, make conjectures about
these values.
i) 30 and 50
ii) 31 and 51
iii) 32 and 52
iv) 33 and 53
J.
Summarize the differences between y = 3x and y = 5x by using their
graphs to determine
i) symmetry
ii) any x- and y-intercepts
iii) when the y-values are increasing
iv) when the y-values are decreasing
v) what happens to the y-values as x gets larger in the positive
direction
vi) what happens to the y-values as x gets larger in the negative
direction
.
Then enter the value in the
home screen and press
MATH
10:49 AM
Reflecting
K.
Will the graph of y = 2x ever touch the x-axis? Explain.
L.
If a is any non-zero base, explain how to write each of the following
in rational form.
i) a0 ii) a1
iii) a2
iv) an
M. In the relations y = x 2 and y = 2x , are the y-values ever negative?
Explain.
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3.6 Exploring Quadratic and Exponential Graphs
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3.6
In Summary
Key Ideas
• Patterns in the table of values for y 2x can be used to determine the meanings
of an and a0 for a Z 0.
• The relations y x2 and y 2x have the following characteristics:
y
y
50
50
y 2x
40
y x2
40
30
30
20
20
10
10
x
x
-8 -6 -4
-2
-10
0
2
4
6
8
The graph is symmetric about
the y-axis.
• The graph has an x-intercept of
0 and a y-intercept of 0.
• The y-values decrease and then
increase as x increases.
• As x increases in the positive
direction, the y-values increase.
• As x increases in the negative
direction, the y-values increase.
•
-8 -6 -4
-2
-10
0
2
4
6
8
The graph is not symmetric.
• The graph has no x-intercept
and a y-intercept of 1.
• The y-values increase as x
increases.
• As x increases in the positive
direction, the y-values increase
much faster than the y-values
for y x 2.
• As x increases in the negative
direction, the y-values decrease
toward 0.
•
Need to Know
• When a non-zero base is raised to the exponent 0, the result is 1: a0 1 for
a Z 0.
• When a non-zero base is raised to a negative exponent, the result is the
1
reciprocal of the base raised to the opposite exponent: an n for a Z 0.
a
FURTHER Your Understanding
1. For part I, you made a conjecture about the value of powers with the
exponent 0. You can confirm your conjecture using exponent rules.
34
a) Express 4 as a single power using the division rule for exponents.
3
NEL
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34
in
34
factored form. Simplify where possible. What is the value of this
expression?
c) Based on your results for parts a) and b), what can you conclude?
53
d) Repeat parts a) to c) using the expression 3 .
5
b) Rewrite the numerator and denominator of the expression
2. For part I, you made conjectures about the values of powers with
negative exponents. These conjectures can also be confirmed using
exponent rules.
33
a) Express
as a single power using the division rule for exponents.
34
33
b) Rewrite the numerator and denominator of the expression in
34
factored form. Simplify where possible. What is the value of this
expression?
c) Based on your results for parts a) and b), what can you conclude?
52
d) Repeat parts a) to c) using the expression 4 .
5
52
e) Repeat parts a) to c) using the expression 5 .
5
3. Evaluate each power. Express your answer in rational form.
c) 80
d) 5-2
a) 2-4
b) 4-1
e) 3-4
f ) 7-2
4. Evaluate each power. Express your answer in rational form.
a) (-2)-5
b) -4-2
c) -70
d) -5-1
e) ( -3)-2
f ) ( -4)-3
5. Evaluate each power. Express your answer in rational form.
1 2
1 -2
2 3
2 -3
3 -2
3 -2
a) a b b) a b
c) a b d) a b
e) - a b
f) a - b
2
2
3
3
4
4
6. Determine the value of n that makes each statement true.
1
8
c) 5n = 1
b) 4n = 64
d) n -3 =
a) 2n =
e) - 3n = 1
27
1
9
f ) ( -n)4 = 16
7. Which do you think is greater: 5-2 or 10-2? Justify your decision.
8. Which do you think is less: (-1)-100 or ( -1)-101? Justify your
decision.
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Chapter Review
FREQUENTLY ASKED Questions
Study
Q:
How do you determine the product of two binomials?
• See Lesson 3.4,
A:
You can use algebra tiles or an area diagram, or you can multiply
symbolically. All three strategies involve the distributive property.
• Try Chapter Review
Aid
Examples 1 to 3.
Questions 13 to 15.
EXAMPLE
Expand and simplify (2x + 3)(2x - 2).
Solution
Using Algebra Tiles
x
Using an Area Diagram
x
x
1
1
1
x2
x2
x
x
x
x
x2
x2
x
⫺1
⫺x
⫺x
⫺1 ⫺1 ⫺1
⫺1
⫺x
⫺x
⫺1 ⫺1 ⫺1
x
x
2x
Multiplying Symbolically
3
(2x + 3)(2x - 2)
2x
⫺2
4x2
⫺4x
6x
⫺6
= 2x(2x - 2) + 3(2x - 2)
= 4x 2 - 4x + 6x - 6
= 4x 2 + 2x - 6
= 4x 2 - 4x + 6x - 6
= 4x 2 + 2x - 6
= 4x 2 - 4x + 6x - 6
= 4x 2 + 2x - 6
Q:
How can you determine whether a quadratic model can be
used to represent data?
A1:
Use the data to create a scatter plot, and draw a curve of good fit.
Confirm that your curve of good fit is a parabola.
A2:
Create a difference table to see if the second differences are
approximately constant.
Q:
How can you determine the equation of a parabola of good
fit in standard form?
A:
NEL
Use the data to create a scatter plot. Estimate the zeros of the
parabola, and then write a general equation in factored form:
y = a(x - r)(x - s). Then substitute the coordinates of a point that
is on, or very close to, the curve of good fit. Substitute the value you
calculated for a into your equation. Expand and simplify your
equation to write it in standard form: y = ax 2 + bx + c.
Study
Aid
• See Lesson 3.4, Example 4,
and Lesson 3.5, Examples 1
to 3.
• Try Chapter Review
Questions 16 to 18.
Chapter 3
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You can check the accuracy of your equation by comparing it with the
equation determined using graphing technology and quadratic
regression. (Note: This only works when the zeros of the curve of
good fit can be estimated or determined.)
Study
Aid
• See Lesson 3.6.
Q:
How are y ⴝ x2 and y ⴝ 2x different?
A:
You can see the differences in their graphs.
• The graph of y = x 2 is a parabola with
y-values that decrease and then increase
as you move from left to right along the
x-axis. The graph of y = 2x is a curve
with y-values that always increase as you
move from left to right along the x-axis.
• The graph of y = x 2 has a vertex and a
minimum value of 0. The graph of
y = 2x has no vertex. It approaches a
minimum value of 0 but will never
equal 0.
• The graph of y = x 2 has an x-intercept
of 0 and a y-intercept of 0. The graph
of y = 2x has no x-intercept and a
y-intercept of 1.
• As x increases in the positive direction,
the y-values for y = 2x increase much
faster than the y-values for y = x 2.
Study
Aid
• See Lesson 3.6.
• Try Chapter Review
Questions 19 and 20.
y
50
40
y ⫽ 2x
30
y ⫽ x2 20
10
x
-10 -6 -2
0
2
6
10
Q:
How do you evaluate a numerical expression that involves
zero or negative exponents?
A:
Any non-zero number raised to the exponent 0 equals 1: a 0 = 1 for
a Z 0.
Any non-zero number raised to a negative exponent equals the
1
reciprocal of the number raised to the opposite exponent: a -n = n
a
for a Z 0.
EXAMPLE
Evaluate.
a) 40
b) 6-2
Solution
a) 40 = 1
184
Chapter Review
1
62
1
=
36
b) 6-2 =
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Chapter Review
PRACTICE Questions
Lesson 3.1
6. Create tables of values for three parabolas that
1. State whether each relation is quadratic. Justify
your decision.
a) y = 4x - 5
b)
7. Use graphing technology to graph the parabola
x
⫺3
⫺2
⫺1
0
1
y
56
35
18
5
⫺4
c) y = 2x(x - 5)
d)
2
3
⫺9 ⫺10
y
6
2
x
-10 -8 -6 -4 -2
0
-2
2
4
6
-4
-6
-8
-10
2. Discuss how the graph of the quadratic relation
y = ax 2 + bx + c changes as a, b, and c are
changed.
8. What does a in the equation y = ax 2 + bx + c
tell you about the parabola?
9. The Rudy Snow Company makes custom
snowboards. The company’s profit can be modelled
with the relation y = - 6x 2 + 42x - 60, where
x is the number of snowboards sold (in thousands)
and y is the profit (in hundreds of thousands of
dollars).
a) How many snowboards does the company
need to sell to break even?
b) How many snowboards does the company
need to sell to maximize their profit?
Lesson 3.3
Lesson 3.2
3. Graph each quadratic relation and determine
i) the equation of the axis of symmetry
ii) the coordinates of the vertex
iii) the y-intercept
iv) the zeros
b) y = x 2 + 2x - 15
4. Verify your results for question 3 using graphing
technology.
5. The x-intercepts of a quadratic relation are - 2
and 5, and the second differences are negative.
a) Is the y-value of the vertex a maximum value
or a minimum value? Explain.
b) Is the y-value of the vertex positive or
negative? Explain.
c) Calculate the x-coordinate of the vertex.
NEL
for each relation below. Then determine
i) the x-intercepts
ii) the equation of the axis of symmetry
iii) the coordinates of the vertex
a) y = - x 2 + 18x
b) y = 6x 2 + 15x
4
a) y = x 2 - 8x
go through the point (2, 7). How do you know
that each table of values represents a parabola?
10. The x-intercepts of a parabola are - 2 and 7,
and the y-intercept is - 28.
a) Determine an equation for the parabola.
b) Determine the coordinates of the vertex.
11. Determine an equation for each parabola.
a) The x-intercepts are 5 and 9, and the
y-coordinate of the vertex is - 2.
b) The x-intercepts are - 3 and 7, and the
y-coordinate of the vertex is 4.
c) The x-intercepts are - 6 and 2, and the
y-intercept is - 9.
d) The vertex is (4, 0), and the y-intercept is 8.
e) The x-intercepts are - 3 and 3, and the
parabola passes through the point (2, 20).
Chapter 3
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12. A bus company usually transports 12 000 people
per day at a ticket price of $1. The company
wants to raise the ticket price. For every $0.10
increase in the ticket price, the number of riders
per day is expected to decrease by 400. Calculate
the ticket price that will maximize revenue.
Lesson 3.4
13. Identify the binomial factors and their products.
a)
x
1 1 1
x
x2
x x x
x
x2
x x x
⫺1
⫺1
⫺1
⫺x
⫺x
⫺x
⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1
b)
⫺6
3x
⫺18x
⫺4
⫺20x
24
a) (x + 5)(x + 4)
d) (4x + 5)(3x - 2)
b) (x - 2)(x - 5)
e) (4x - 2y)(5x + 3y)
c) (2x - 3)(2x + 3) f ) (6x - 2)(5x + 7)
15. Expand and simplify.
a) (2x + 6)2
b) -2( - 2x + 5)(3x + 4)
c) 2x(4x - y)(4x + y)
1
2
3
4
5
6
Height (m)
0.0 25.1 40.4 45.9 41.6 27.5 3.6
a) Sketch a curve of good fit.
b) Is the curve of good fit a parabola? Explain.
c) Determine the equation of your curve of good
fit. Express your answer in standard form.
d) Estimate the height of the rocket after 4.5 s.
e) When is the rocket at a height of 20 m?
0
2
4
6
8
10
Height (m) 1200 1180 1120 1020
880
700
a) Draw a scatter plot of the data.
b) Sketch a curve of good fit.
c) Is the curve of good fit a parabola? Explain.
d) Determine the equation of your curve of good
fit. Express your answer in standard form.
e) Estimate the time when the sandbag hits
the water.
19. Evaluate. Express your answers in rational form.
your answer in standard form.
y
a) 2-3
b) -5-1
2 -2
c) a b
5
d) (-9)0
e) 4-3
1 -2
f) - a b
6
2
20. Which do you think is greater: a b or 3-2 ?
1
4
8
Justify your decision.
6
4
21. For what postive values of x is x 2 greater than 2x ?
2
x
186
0
Lesson 3.6
16. Determine the equation of the parabola. Express
0
-2
Time (s)
Time (s)
14. Expand and simplify.
-2
The table shows its height, y, in metres after
x seconds.
air balloon to make the balloon rise. The table
shows the height of the sandbag at different
times as it falls.
15x2
-6 -4
17. A model rocket is shot straight up into the air.
18. A sandbag is dropped into the ocean from a hot
5x
10
Lesson 3.5
2
Chapter Review
4
6
8
How do you know?
10
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Chapter Self-Test
1. State the zeros, vertex, and equation of the axis of symmetry
y
of the parabola at the right.
6
2. The points (⫺9, 0) and (19, 0) lie on a parabola.
4
a) Determine an equation for its axis of symmetry.
b) The y-coordinate of the vertex is ⫺28. Determine an equation
for the parabola in factored form.
c) Write your equation for part b) in standard form.
2
x
-8
-6 -4
x ⫺1
0
y
2 ⫺3 ⫺14 ⫺31
1
1
2
3
y ⫺4 ⫺3
0
3
4
5
12
4. Sketch each graph. Label the intercepts and the vertex using
their coordinates.
a) y = (x - 6)(x + 2)
-2
2
4
-4
3. Decide, without graphing, whether each data set can be modelled by
a quadratic relation. Explain how you made your decision.
a)
b) x 0 1 2
-2
0
Process
Checklist
✔ Question 2: Did you relate
b) y = - (x - 6)(x + 4)
the characteristics of the
graphical representation
of the relation with its
equation?
5. The population, P, of a city is modelled by the equation
P = 14t 2 + 820t + 42 000, where t is the time in years.
When t = 0, the year is 2008.
a) Determine the population in 2018.
b) When was the population about 30 000?
✔ Questions 5 and 7: Did you
select appropriate problem
solving strategies for each
situation?
6. Expand and simplify.
c) -5(x - 4)2
a) (2x - 3)(5x + 2) b) (3x - 4y)(5x + 2y)
7. A toy rocket is placed on a tower and launched straight up. The table
shows its height, y, in metres above the ground after x seconds.
Time, x (s)
0
1
2
3
4
5
6
7
8
Height, y (m)
16
49
72
85
88
81
64
37
0
✔ Question 8: Did you
make connections to
communicate a variety
of ways to relate modelling
with linear and quadratic
relations?
a) What is the height of the tower?
b) How long is the rocket in flight?
c) Do the data in the table represent a quadratic relation? Explain.
d) Create a scatter plot. Then draw a curve of good fit.
e) Determine the equation of your curve of good fit.
f ) What is the maximum height of the rocket?
8. In what ways is modelling a problem using a quadratic relation similar
to using a linear relation? In what ways is it different?
9. Evaluate.
a) 7 - 2
NEL
b) - 30
2 -4
c) - a b
3
d) -5 - 3
Chapter 3
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Chapter Task
Comparing the Force of Gravity
Gravity is the force of attraction between two objects. This force varies in
our solar system. For example, the Moon’s diameter is about one-fourth of
Earth’s diameter, so the Moon’s gravity is much less than Earth’s.
A measure of the strength of gravity is the value g, which is the acceleration
(or rate of change of velocity) of a freely falling object. One strategy for
calculating g is to drop an object and time its fall to the ground. These
tables show the time that an object takes to fall from a height of 10 m
on both Earth and the Moon.
Earth
Time (s)
0.0
Height (m)
0.2
0.4
0.6
0.8
10.0000 9.8038 9.2152
8.2342
Time (s)
0.0
Height (m)
10.0000 9.7975 9.1900
1.0
1.2
1.4
6.8608 5.0950
2.9368
0.3862
1.5
2.0
3.0
3.5
8.1775
6.7600 4.9375
2.7100
0.0775
Moon
Task
Checklist
✔ Did you label your graphs?
A.
Create a scatter plot for each data set and draw a curve of good fit.
B.
Do the data show that the relation between time and height
is quadratic on Earth or the Moon? Explain.
C.
Estimate the location of the vertex, the axis of symmetry, and the zeros.
D.
Use a strategy of your choice to determine the equations of curves
of good fit in standard form.
g
An equation that models this situation is y = H - a b x 2, where
2
H is the initial height of the object above the ground and y is the
E.
height of the object at time x. If x is measured in seconds, and y and H
✔ Did you explain your
are measured in metres, then the units for g are metres per square
thinking clearly?
second (m/s2). Use this information and the equations you determined
✔ Did you calculate the factor
Chapter Task
2.5
What factor relates the values of g on the Moon and Earth?
calculations?
188
1.0
?
✔ Did you include your
that relates the value of g
on the Moon to the value
of g on Earth?
0.5
for part D to calculate the values of g on Earth and the Moon.
F.
Calculate the factor that relates the value of g on the Moon to
the value of g on Earth.
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Chapters
Chapters
1–3
Cumulative Review
Multiple Choice
1. Which ordered pair satisfies both
6. The music students at a school sold 771 tickets
3x - 2y = - 11 and 5x + y = - 1?
C. (1, 4)
D. (1, 6)
A. (5, 2)
B. (1, 7)
2. Which linear system has the solution (5, 2)?
A. y = x - 3
C. x + y - 7 = 0
2x + y = 8
B. x - 2y = 1
3x - 4y = 7
4y = x + 5
D. -3x + 5y = - 5
4x - y = 11
3. Which equation is equivalent to
5x - 4y + 6 = 0?
A. y = 1.25x + 1.5 C. y = - 1.5 - 1.25x
B. x = 0.8y + 1.2 D. x = - 1.25y - 0.83
4. Hannah pays a one-time registration charge and
regular monthly fees to belong to a fitness club.
After three months, she has paid $330. After
eight months, she has paid $655. What is the
registration charge, and what is the monthly fee?
A. $110, $81.88
C. $65, $135
B. $263, $67
D. $135, $65
5. Which linear system is equivalent to the linear
system shown in the graph?
A. 2x + y = 9
C. x - y = 3
4x - y = 3
6x = 12
B. 6x + 2y = 12
D. 3y = - 3
2x - 2y = 6
8x + y = 12
y
4x y 9
2x y 3 4
2
x
-6 -4
-2
0
-2
-4
-6
-8
2
4
6
8
to their spring concert, for a total of $6302.
Students paid $6 for a ticket, and non-students
paid $10. How many non-students attended
the concert?
A. 352
C. 630
B. 419
D. 141
7. On weekends, Brad likes to go cycling. He
cycles partly along trails and partly off-trail,
through hilly wooded areas. He cycles at
20 km/h on trails and at 12 km/h off-trail.
One day, he cycled 48 km in 3 h. How far
did he cycle off-trail?
A. 18 km
C. 15 km
B. 24 km
D. 30 km
8. Which line segment has the midpoint (3, 2)?
A. AB; A(2, 1), B(4, 3)
B. CD; C(4, 2), D(1, 3)
C. EF; E(1, 3), F(7, 7)
D. GH; G(2, 2), H(8, 6)
9. A triangle has vertices at A(2, 7), B(2, 1), and
C(8, 3). Which equation represents the
median from vertex A?
A. y = - 6x + 19
C. y = 8x - 9
B. y = - 3x + 13
D. y = - 8x + 23
10. Which point is closest to P(1, 4)?
A. Q(2, 5)
C. S(3, 1)
B. R(1, 2)
D. T(4, 6)
11. Which point is closest to the line 3x + 2y = 6?
A. (0, 0)
C. (1, 1)
B. (1, 3)
D. (2, 4)
12. Determine the equation of a circle that has a
diameter with endpoints ( - 2, 7) and (2, 7).
A. x 2 + y 2 = 14
C. x 2 + y 2 = 45
2
2
B. x + y = 25
D. x 2 + y 2 = 53
-10
NEL
Chapters 1–3
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13. Which triangle is a right triangle?
A. ^ ABC ; A(3, 3), B(2, 5), C(6, 5)
B. ^ DEF ; D(2, 2), E(0, 6), F(5, 1)
C. ^GHI; G(5, 1), H(0, 2), I(1, 6)
D. ^ JKL; J(2, 6), K(2, 5), L(1, 6)
14. A quadrilateral has vertices at A(4, 2),
B(5, 2), C(3, 4), and D(8, 1). Which type
of quadrilateral is it?
A. parallelogram
C. rectangle
B. rhombus
D. trapezoid
15. This arch is formed by an arc of a circle. What is
the radius of the circle?
0.750 m
3.000 m
A. 1.500 m
B. 1.875 m
x-intercepts at (2, 0) and (4, 0). The second
differences for the quadratic relation are
negative. Which statement about the quadratic
relation is true?
A. It has a maximum value, which is positive.
B. It has a minimum value, which is positive.
C. It has a maximum value, which is negative.
D. It has a minimum value, which is negative.
17. The points (2, 7) and (4, 7) lie on the
parabola defined by the equation
y = 3x 2 - 6x - 17. What are the coordinates
of the vertex of the parabola?
A. (1, 7)
C. (6, 17)
B. (1, 20)
D. (1, 17)
18. A parabola has zeros at (1, 0) and (5, 0),
and passes through (0, –5). Which equation
describes the parabola?
A. y = (x - 1)(x + 5)
B. y = (x + 1)(x - 5)
C. y = (x + 1)(x + 5)
D. y = (x - 1)(x - 5)
Cumulative Review
customized logos. The company’s annual profit,
P, in thousands of dollars, is modelled by
P = ( -6x + 78)(x + 3), where x represents
the number of dozens of T-shirts produced, in
thousands. What is the company’s maximum
annual profit?
A. $60 000
C. $384 000
B. $13 000
D. $234 000
20. A quadratic relation has an equation of the form
y = a(x - r)(x - s). The graph of the relation
has zeros at x = - 3 and x = 7, and passes
through (5, 24). What is the value of a?
2
3
A. C. 3
2
3
B.
D. 24
2
21. A quadratic relation has zeros at x = 16
C. 3.000 m
D. 3.750 m
16. The graph of a quadratic relation has
190
19. Tommy’s Custom T’s produces T-shirts with
and x = 28, and passes through (32, 16).
Which equation describes the relation?
A. y = (x - 16)(x - 28)
B. y = 0.25(x - 16)(x + 28)
C. y = 0.5(x - 16)(x - 28)
D. y = 0.25(x - 16)(x - 28)
22. Which expression is the product
of (3x - 4) and (7x + 6)?
A. 21x 2 - 10x - 24
B. 10x 2 - 2x - 24
C. 21x 2 - 46x - 24
D. 21x 2 - 10x + 24
23. Which number is equivalent to a b
16
81
81
B.
16
A.
4
9
16
C. 81
3
D.
2
-2
?
24. Which number is equivalent to 50?
A. 5
C. 1
B. 0
D. 1
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Cumulative Review
Investigations
Home Heating Economics
Jenny and Oliver are building a new home. They have researched different
types of heating systems they could install. The costs of three heating
systems are given as follows:
• gas furnace: $4000 to install and $1250/year to operate
• electric baseboard heaters: $1500 to install and $1000/year to operate
• geothermal heat pump: $12 000 to install and $400/year to operate
Lung Cancer Data
Year
Number of Cases
per 100 000
Population
1976
75.7
1977
78.6
1978
85.1
1979
83.9
1980
83.2
Cyclic Quadrilaterals
1981
91.2
In any triangle, you can draw a circle that passes through all three vertices.
Can you draw a circle that passes through all four vertices of a quadrilateral?
Consider the quadrilateral with vertices A(4, 10), B(3, 9), C(9, 1), and
D(12, 6).
1982
92.6
1983
95.2
1984
97.1
1985
93.2
1986
96.4
1987
95.0
1988
95.5
1989
93.6
1990
92.7
1991
90.7
1992
90.1
1993
91.3
1994
86.7
1995
84.4
1996
82.0
1997
78.9
1998
79.9
1999
79.2
2000
75.6
25. a) Use equations and a graph to show how the total cost varies over
time for each heating system.
b) Determine any intersection points. State the conditions under
which each heating system costs less than one or both of the others.
c) Which heating system would you recommend? Justify your choice.
26. a) Draw a diagram of quadrilateral ABCD. How could you use analytic
geometry to determine whether or not a circle that passes through A,
B, C, and D can be drawn? If it can, the quadrilateral is cyclic.
b) Carry out the plan you described for part a). Is quadrilateral
ABCD cyclic?
c) Draw the diagonals of quadrilateral ABCD, and determine their
point of intersection, E. Show that AE * EC = BE * ED.
d) What types of quadrilaterals could be cyclic?
Lung Cancer Rates for Canadian Males
The table at the right gives the lung cancer rates for Canadian males per
100 000 population.
27. a) Using 0 for 1976, construct a scatter plot and curve of good fit.
b) Without using quadratic regression, determine an equation
for your curve of good fit.
c) Using quadratic regression, determine an equation for the curve
of best fit. Compare your equations for parts b) and c).
d) What do you expect to happen to the lung cancer rates
for Canadian males past the year 2000? Explain.
NEL
Chapters 1–3
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Chapter
4
Factoring
Algebraic
Expressions
GOALS
You will be able to
• Determine the greatest common factor
in an algebraic expression and use it
to write the expression as a product
• Recognize different types of quadratic
expressions and use appropriate
strategies to factor them
? Police detectives must often
retrace a suspect’s movements,
step by step, to solve a crime.
How can working backwards help
you determine the value of each
symbol?
(䉱x ⴙ 䊉)(䊏x ⴙ 䉬) ⴝ 3x 2 ⴙ 11x ⴙ 10
NEL
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Getting Started
WORDS YOU NEED to Know
1. Match each word with the mathematical expression that best illustrates
its definition.
a) binomial
b) coefficient
c) factoring
d) monomial
e) variable
f ) trinomial
i) 2x 5
ii) 4(2x 3) 8x 12
iii) 7x 2 3x 1
iv) 8x 2
g) expanding
h) like terms
v) 24 2 2 2 3
vi) 4xy
vii) x
viii) 6y and 8y
SKILLS AND CONCEPTS You Need
Simplifying an Algebraic Expression
Study
Aid
• For more help and practice,
see Appendix A-8.
To simplify an algebraic expression, create a simpler equivalent expression
by collecting like terms.
EXAMPLE
Simplify (4x2 2) (2x2 3).
Solution
Using Symbols
Using an Algebra Tile Model
(4x2 2) (2x2 3)
(4x2 2) (4x
2(
(2x
3) 2 3)
2x
2)2 4x2 2 2x2 3
4x2 2x2 2 3
2x2 5
x
x
x
x
x
x
x
x
1
1
x
x
x
x
1
1
1
1
1
1
1
1
2x2 5
2. Simplify each expression.
a) 4x - 6y + 8y - 5x
b) 5ab - 6a 2 + 6ab - 3a 2 - 11ab + 9b 2
c) (2x - 5y) + (7x + 4) - (5x - y)
d) (7a - 2ab) - (4b + 5a) + (ab - 3a)
194
Getting Started
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Getting Started
Expanding an Algebraic Expression
To expand an algebraic expression, multiply all parts of the expression
inside the brackets by the appropriate factor. You can use the distributive
property algebraically or geometrically with a model.
Multiplying by a Monomial
Study
Aid
• For more help and practice,
see Appendix A-8.
Multiplying by a Binomial
(a b)(c d) a(c d) b(c d)
a(b c) ab ac
ac ad bc bd
EXAMPLE
Expand and simplify.
a) 2(2x 4)
b) (x 1)(3x 2)
Solution
a)
Distributive
Property
Area Model
x
2(2x 4)
4x 8
x
b)
1 1 1 1
1
1
Distributive Property
Algebra Tile Model
(x 1)(3x 2)
1
x
x
x
1 1
x
x2
x2
x2
x x
x
x
x
1 1
x(3x 2) 1(3x 2)
3x2 2x 3x 2
3x2 x 2
The area is 4x 8.
The area is 3x2 x 2.
3. Expand and simplify.
a) 7(2x - 5)
b) - 5x(3x 2 - 4x + 5)
c) 2(4x 2 + 3x + 1) - 2x(8x - 3)
d) (d - 6)(d + 2)
e) (3a - 7b)(4a - 3b)
f ) 6x(2x + 1)2
4. Determine the multiplication expression and product for each model.
a)
x
x
1 1 1 1 1
c)
1
1
1
1
b)
x
x
x
1 1
d)
1
1
x
x
x
x
x
x
1 1
1 1
x
x2
x2
x2
x x
x
x
x
1 1
1
1
x
x
x
x
x
x
1 1
1 1
x
x2
x2
x2
x x
x
x
x
1 1
x
NEL
Chapter 4
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Aid
PRACTICE
• For help, see the Review
5. Simplify.
Study
a) (x 5)(x 7)
b) (-6a 2)(3a 4)(2a)
of Essential Skills and
Knowledge Appendix.
Question
Appendix
5
A-3
c) (4y)(3y 2) ÷ 2y 3
d) 20z 5 ÷ (-4z 3)( -z 2)
6. Create an expression for each description.
a) a monomial with a coefficient of 5
b) a binomial with coefficients that are even numbers
c) a trinomial with coefficients that are three consecutive numbers
d) a quadratic trinomial
7. Identify the greatest common factor for each pair.
a) 28, 35
b) 36, 63
e) 25, 5x 2
f ) 12y, 6x
c) 99, 90
d) 4x, 8
8. a) Determine the x-intercepts, the equation of the axis of symmetry,
and the vertex of y = (x - 4)(x + 8).
b) Use the information you determined for part a) to sketch
this parabola.
c) Express the equation in standard form.
9. Sketch algebra tiles, like those shown at the left, to represent
x2
x2
x x
1 1
each expression.
a) 3x 2
b) 2x 4
c) 2x 2 - x + 3
d) -2x 2 - 1
e) 2x 2 - x - 1
f ) 1 + 2x - 3x 2
10. Match each expression with the correct diagram.
a) x 3
i)
b) 3x 2
1
ii)
x
x
c) x 2 - 1
x
1 1
iii)
x
1 1 1
x2
11. The area model shown below represents x 2 + 3x.
x
1 1 1
x
Sketch an area model to show each expression.
a) 2x 2 - 3x
c) -x 2 - 3x
b) 3x 6
d) 2x 2 + x
12. Decide whether you agree or disagree with each statement. Explain why.
a) All even numbers contain the number 2 in their prime factorization.
b) 6x 2y 3 can be written as a product of two or more algebraic
expressions.
c) The only way to factor 100 is 10 10.
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Getting Started
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Getting Started
APPLYING What You Know
Designing a Geometric Painting
Sara and Josh are creating a large square painting for their school’s Art Fair.
They are planning to start with a large rectangle, measuring 108 cm by
144 cm. They will divide the rectangle into congruent squares, which
they will paint different colours. They want these squares to be as large
as possible. To create the final painting, they will use copies of the 108 cm
by 144 cm rectangle to create a large square with the least possible whole
number dimensions.
144 cm
108 cm
?
What are the dimensions of the small squares and the final
large square painting?
A.
How do you know that the side length of the small squares, inside the
108 cm by 144 cm rectangle, cannot be 8 cm?
B.
Why must the side length of the small squares be a factor of 108 and 144?
C.
What is the side length of the largest square that can be used to divide
the 108 cm by 144 cm rectangle?
D.
Why does the side length of the final large square painting (created
using copies of the 108 cm by 144 cm rectangle) have to be a multiple
of both 108 and 144?
E.
What is the side length of the smallest final square painting that can be
created using copies of the 108 cm by 144 cm rectangle?
NEL
Chapter 4
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Common Factors in Polynomials
GOAL
• algebra tiles
Factor algebraic expressions by dividing out the greatest common
factor.
LEARN ABOUT the Math
Yasmine squared the numbers 5 and 6, and then added 1 to get a sum of 62.
52 + 62 + 1 = 25 + 36 + 1
= 62
She repeated this process with the numbers 8 and 9, and got a sum of 146.
82 + 92 + 1 = 64 + 81 + 1
= 146
Both 62 and 146 are divisible by 2.
146
62
= 31 and
= 73
2
2
?
Is a number that is 1 greater than the sum of the squares
of two consecutive integers always divisible by 2?
EXAMPLE 1
Selecting a strategy to determine
the greatest common factor
Lisa’s Solution: Selecting an algebraic strategy
n2 + (n + 1)2 + 1
= n 2 + (n + 1)(n + 1) + 1
= n 2 + (n 2 + 2n + 1) + 1
= 2n 2 + 2n + 2
= 2(n 2 + n + 1)
Communication
Tip
GCF is an abbreviation for
Greatest Common Factor.
198
2 is always a factor of this expression.
Therefore, 1 greater than the sum
of the squares of two consecutive
integers is always divisible by 2.
4.1 Common Factors in Polynomials
I let n represent the first integer.
Then two consecutive integers
are n and n 1. I wrote an
expression for the sum of their
squares and added 1.
I simplified the expression by
expanding and collecting like
terms.
I factored the algebraic
expression by determining the
GCF of its terms. The GCF is 2.
I wrote an equivalent expression
as a product.
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4.1
Abdul’s Solution: Representing with an area model
x2
x
1
x2
x
I used algebra tiles to show x2 and
(x 1)2. Then I added one unit tile.
I wanted to see if I could make
two equal groups. If I could, this
would show that the number is
divisible by 2.
1
x2
x2
x
I moved tiles around to get
two equal groups.
x
1
1
Since there are two equal
groups, the total value is always
a multiple of 2.
The result will always be
divisible by 2.
Reflecting
A.
Why did both Lisa and Abdul use a variable to prove that the result
is always even?
B.
Why did Lisa need to factor in order to solve this problem?
C.
Which strategy do you prefer? Explain why.
to express a number as the
product of two or more
numbers, or express an algebraic
expression as the product of two
or more terms
APPLY the Math
EXAMPLE 2
Selecting a strategy to factor a polynomial
Factor 3x 2 - 6x over the set of integers.
Communication
Noel’s Solution: Representing with an area model
x
x2
x2
x2
3x 6
x2
x2
x2
x
x
x
x
3x
x
x
x2
NEL
factor
x x x x x x
I used algebra tiles
to represent each
term. I used red
tiles for positive
terms and blue tiles
for negative terms.
I arranged these
tiles to form two
different rectangles.
Tip
Factoring over the set of
integers means that all the
numbers used in each factor
of an expression must be
integers. When you are asked
to factor, this is implied.
Chapter 4
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3x 2 - 6x = x(3x - 6)
and
3x 2 - 6x = 3x(x - 2)
3x 2 - 6x = x(3x - 6)
= x(3)(x - 2)
= 3x(x - 2)
Communication
Tip
An algebraic expression is
factored fully when only 1 or
1 remains as a factor of
every term in the factorization.
When you are asked to factor,
you are expected to factor
fully.
The polynomial is factored
fully since one of the factors
is the greatest common
factor of the polynomial.
The dimensions of each rectangle
are factors of the expression.
I noticed that x(3x 6) could be
factored again, since 3x 6
contains a common factor of 3.
The greatest common factor is
3x, which is the first factor in
3x(x 2).
Connie’s Solution: Reasoning symbolically
3x 2 - 6x
= 3(x 2) + 3(-2x)
To factor, I need to determine
the greatest common factor.
The GCF of the coefficients
3 and 6 is 3.
= 3x(x) + 3x(-2)
The GCF of the variable parts
x2 and x is x.
The greatest common factor of
both terms in 3x 2 - 6x is 3x.
3x 2 - 6x = 3x(x - 2)
I used the distributive property
to write an equivalent expression
as the product of the factors.
EXAMPLE 3
Connecting the distributive property
with factoring
Factor.
a) 10a 3 - 25a 2
b) 9x 4y 4 + 12x 3y 2 - 6x 2y 3
Antonio’s Solution
a) The greatest common factor
of the terms in 10a 3 - 25a 2
is 5a2.
10a 3 - 25a 2 = 5a 2(2a - 5)
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4.1 Common Factors in Polynomials
The GCF of the coefficients 10
and 25 is 5. The GCF of the
variable parts a3 and a2 is a2.
I used the distributive property to
write an equivalent expression as
the product of the factors.
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4.1
b) The greatest common factor
of the terms in
9x 4y 4 + 12x 3y 2 - 6x 2y 3
is 3x 2y 2.
9x 4y 4 + 12x 3y 2 - 6x 2y 3
= 3x 2y 2(3x 2y 2 + 4x - 2y)
EXAMPLE 4
The GCF of the coefficients 9,
12, and 6 is 3. The GCF of the
variable parts is x2y2.
I used the distributive property
to write an equivalent expression
as the product of the factors.
Reasoning to factor a polynomial
Factor each expression.
a) 5x(x - 2) - 3(x - 2)
b) ax - ay - 5x + 5y
Joanne’s Solution
a) 5x(x - 2) - 3(x - 2)
Both 5x and 3 are multiplied
by x 2, so x 2 is a common
factor.
= (x - 2)(5x - 3)
I wrote an equivalent expression
using the distributive property.
b) ax - ay - 5x + 5y
NEL
I noticed that the first two terms
contain a common factor of a
and the last two terms contain
a common factor of 5.
To factor the expression, I used
a grouping strategy. I grouped
the terms that have the same
common factor.
= a(x - y) - 5(x - y)
I wrote an equivalent expression
using the distributive property.
Both a and 5 are multiplied by
x y, so x y is a common factor.
Since this binomial is the same in
both terms, I can factor further.
= (x - y)(a - 5)
I wrote an equivalent expression
using the distributive property.
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In Summary
Key Ideas
expanding
• Factoring is the opposite of expanding. Expanding involves multiplying,
and factoring involves looking for values to multiply.
• One way to factor an algebraic expression is to look for the greatest
common factor of the terms in the expression. For example,
5x2 10x 15 can be factored as 5(x2 2x 3) since 5 is the
greatest common factor of all the terms.
4x(2x 3) 8x2 12x
factoring
Need to Know
• It is possible to factor an algebraic expression by dividing by a common
factor that is not the greatest common factor. This will result in another
expression that still has a common factor. For example,
8x 16 4(2x 4)
4(2)(x 2)
8(x 2)
• An algebraic expression is factored fully when only 1 or 1 remains as
a factor of every term in the factorization. In the example above, 8x 16
is factored fully when it is written as 8(x 2) or 8(x 2).
• A common factor can have any number of terms. For example, a
common factor of the terms in 9x2 6x is 3x, which is a monomial.
A common factor of the terms in (3x 2)2 4(3x 2) is (3x 2),
which is a binomial.
CHECK Your Understanding
1. In each algebra tile model below, two terms in an algebraic expression
have been rearranged to show a common factor.
i) Identify the algebraic expression, and name the common factor
that is shown.
ii) Determine the greatest common factor of each expression.
a)
202
b)
x2
x2
x x x x x x
x2
x2
x x x x
x2
x2
x x x x x x
x2
x2
x x x x
x2
x2
x x x x
4.1 Common Factors in Polynomials
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4.1
2. Each area diagram represents a polynomial.
i) Identify the polynomial, and determine a common factor
of its terms.
ii) Determine any other common factors of the terms
in the polynomial.
a)
x
1 1 1 1
b)
1
1
1
1
x
x
x
1 1 1 1
x
x
3. State the GCF of each pair of terms.
a) 6x and 10x
b) 15a 3 and 20a 2
c) ab and a 2b 2
d) -2x 4y 4 and 8x 3y 5
PRACTISING
4. Identify the greatest common factor of the terms in each expression.
a) 3x 2 - 9x + 12
b) 5x 2 + 3x
c) x 2y - xy 2
d) 4x(x - 1) + 3(x - 1)
5. Determine the missing factor.
a) 8xy = (䊏)(2xy)
b) - 6x 2 = (3x 2)(䊏)
c) 15x 4z = (5x 2)(䊏)
d) -49a 2b 5 = (䊏)(7b 3)
e) -12x 3y 3 = (3y 3)(䊏)
f ) 30m 2n 3 = ( -5m 2n)(䊏)
6. Determine the missing factor.
a) 4x - 4y = (䊏)(x - y) d) 36x 2 - 32y 3 = (4)(䊏)
b) 8x - 2y = (2)(䊏)
e) -24x 2 - 6y = (䊏)(4x 2 + y)
c) 5a + 10b = (䊏)(a + 2b) f ) 45a 4 - 54a 3 = (9a 3)(䊏)
7. Determine the greatest common factor of each expression.
a) 7x 2 + 14x - 21
b) 3b 2 + 15b
c) 12c 2 - 8c + 16
d) -25m 2 - 10m
e) 3d 4 - 9d 2 + 15d 3
f) y 3 + y 5 - y 2
8. Factor each expression. Then choose one expression, and describe
the strategy you used to factor it.
a) 9x 2 - 6x + 18
d) 2b(b + 4) + 5(b + 4)
b) 25a 2 - 20a
e) 4c(c - 3) - 5(c - 3)
3
4
c) 27y - 9y
f ) x(3x - 5) + (3x - 5)(x + 1)
9. Factor each polynomial. Then identify the two polynomials that have
the same trinomial as one of their factors.
a) dc 2 - 2acd + 3a 2d
d) 2a 2c 4 - 4a 3c 3 + 6a 4c 2
2
3
b) –10a c + 20ac - 5ac
e) 3a 5c 3 - 2ac 2 + 7ac
c) 10ac 2 - 15a 2c + 25
f ) 10c 3d - 8cd 2 + 2cd
NEL
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10. Factor each expression.
a) ax - ay + bx - by
b) 10x 2 + 5x - 6xy - 3y
c) 3mx + 3my + 2x + 2y
4b 1
d) 5my + tm + 5ny + tn
e) 5wx - 10w - 3tx + 6t
f ) 4mnt - 16mn - t + 4
11. The area of the trapezoid at the left is A = 70b + 10. Determine
the height.
12. Examine each quadratic relation below.
?
i) Express the relation in factored form.
ii) Determine the zeros and the equation of the axis of symmetry.
iii) Determine the coordinates of the vertex.
iv) Sketch the graph of the relation.
a) y = 2x 2 - 10x
b) y = - x 2 - 8x
3b
13. Determine an expression, in factored form, that can be used
to determine the surface area of any rectangular prism.
14. Two parabolas are defined by y = 10x - x 2 and y = 3x 2 - 30x.
A
What is the distance between their maximum and minimum values?
15. a) Write three quadratic binomials whose greatest common factor
K
is 5x. Then factor each binomial.
b) Write three quadratic trinomials whose greatest common factor
is 3x. Then factor each trinomial.
16. Marek says that the greatest common factor of -5x 3 + 10x 2 - 20x
C
is 5x. Jen says that the greatest common factor is -5x. Explain why
both Marek and Jen are correct.
17. Show that 1 greater than the sum of the squares of any three
T
consecutive integers is always divisible by 3.
18. Once you have factored an algebraic expression, how can you check
to ensure that you have factored correctly? Explain why your strategy
will always work.
Extending
19. Determine the expression, in factored form, that represents the shaded
area between the circle and the square in the diagram at the left.
r
r
20. Factor the numerator in each expression, and then simplify
the expression. Assume that no variable equals zero.
2x 2y + 3xy 2
-12x 3y 2 - 18x 2y 3
a)
c)
xy
6x 2y 2
6x 3y + 12x 3y 2
3x 4 + 6x 3 + 9x 2
b)
d)
6x 3y
3x 2
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Exploring the Factorization
of Trinomials
GOAL
Discover the relationship between the coefficients and constants
in a trinomial and the coefficients and constants in its factors.
EXPLORE the Math
YOU WILL NEED
• algebra tiles
x2
1
1
1
x x x x
A trinomial can be represented using algebra tiles. If you can arrange
the tiles as a rectangle, then the expression represents the area of the
rectangle. The dimensions of the rectangle represent its factors.
For example, to factor x 2 + 4x + 3, arrange one x 2 tile,
four x tiles, and three unit tiles into a rectangle. The diagram
shows that x 2 + 4x + 3 (x 1)(x 3).
1
x
1
?
What is the relationship between the terms in a trinomial and
the terms in its factors?
x
x2
x x x
Factor each trinomial using algebra tiles. Sketch your model, and
record the two factors.
i) x 2 + 2x + 1
iii) x 2 + 6x + 8
v) x 2 - 4x + 3
ii) x 2 + 5x + 6
iv) x 2 - 2x + 1
vi) x 2 - 3x + 2
x
1
A.
B.
Factor each trinomial using algebra tiles. Sketch your model, and
record the two factors.
i) x 2 - 2x - 3
iii) x 2 - 2x - 8
v) x 2 - 3x - 10
ii) x 2 + 3x - 4
iv) x 2 + x - 6
vi) x 2 + 4x - 5
C.
If the coefficient of x 2 in a trinomial is a value other than 1, it may be
possible to factor it using rectangular arrangements of algebra tiles.
Factor each trinomial using algebra tiles. Sketch your model, and
record the two factors.
i) 2x 2 + 3x + 1
iii) 3x 2 - 4x + 1
v) 2x 2 - 7x - 4
ii) 2x 2 + 5x + 2
iv) 2x 2 - 7x + 6
vi) 3x 2 + 5x - 2
D.
Examine the trinomials in parts A, B, and C.
i) Compare the coefficient of x 2 in each trinomial with the
coefficients of x in the factors. What do you notice?
ii) Compare the constant term in each trinomial with the constant
terms in the factors. What do you notice?
iii) Compare the coefficient of x in each trinomial with the coefficients
and constants in the factors. What do you notice?
NEL
x2 4x 3
1
1
1
1
(x 1)(x 3)
Communication
Tip
When negative tiles are used,
add tiles to make a rectangle
if necessary. To do this without
changing the value of the
trinomial, add positive and
negative tiles of equal value.
Chapter 4
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Reflecting
E.
What type of polynomial are the factors of trinomials of the form
ax 2 + bx + c?
F.
How can you use the values of a, b, and c in ax 2 + bx + c to help
you factor a trinomial, without using an algebra tile model?
G.
Can a rectangular arrangement of algebra tiles be created for
x 2 + 3x + 1 or 2x 2 + x + 1? What does this imply?
In Summary
Key Idea
• To factor a trinomial of the form ax 2 + bx + c using algebra tiles,
you need to form a rectangle. The factors are the dimensions
of the rectangle.
Need to Know
• A factorable trinomial of the form ax 2 + bx + c may have
two binomials as its factors.
• If a rectangle cannot be created for a given trinomial, the trinomial
cannot be factored.
• When factoring a trinomial using algebra tiles, you may need to add
tiles to create a rectangular model. This requires you to add positive and
negative tiles of equal value.
FURTHER Your Understanding
1. Use algebra tiles to factor each polynomial. Sketch your model, and
record the two factors.
a) x 2 + 7x + 12
b) x 2 - x - 12
c) x 2 - 4x + 4
d) x 2 - x - 6
e) x 2 - 5x + 6
f) x 2 - 4
2. Check your results for each expression in question 1 by expanding.
3. Use algebra tiles to factor each polynomial. Sketch your model, and
record the two factors.
a) 2x 2 + 7x + 3
b) 4x 2 + 4x - 3
c) 4x 2 - 4x + 1
d) 3x 2 + 7x - 6
e) 2x 2 - 9x + 4
f ) 6x 2 + 7x + 2
4. Check your results for each expression in question 3 by expanding.
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4.2 Exploring the Factorization of Trinomials
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Factoring Quadratics: x 2 ⴙ bx ⴙ c
GOAL
YOU WILL NEED
Factor quadratic expressions of the form ax 2 ⫹ bx ⫹ c, where a ⫽ 1.
• algebra tiles
INVESTIGATE the Math
Brigitte remembered that an area model can be used to multiply
two binomials. To multiply (x ⫹ r)(x ⫹ s), she created the model
at the right and determined that the product is quadratic.
?
How can an area model be used to determine the factors
of a quadratic expression?
A.
Use algebra tiles to build rectangles with the dimensions shown in
the table below. Copy and complete the table, recording the area
in the form x 2 + bx + c.
Length
Width
x⫹3
x⫹4
x⫹3
x⫹5
x⫹3
x⫹6
x⫹4
x⫹4
x⫹4
x⫹5
Area: x2 ⴙ bx ⴙ c
Value of b
Look for a pattern in the table for part A. Use this pattern to predict
the length and width of a rectangle with each of the following areas.
i) x 2 + 8x + 12
iii) x 2 + 11x + 30
ii) x 2 + 10x + 21
iv) x 2 + 11x + 18
C.
The length of a rectangle is x + 4, and the width is x - 3.
What is the area of the rectangle?
D.
Repeat part A for rectangles with the following dimensions.
NEL
Width
x⫺3
x⫹5
x⫹3
x⫺6
x⫺2
x⫺2
x⫺1
x⫺5
Area: x2 ⴙ bx ⴙ c
Value of b
⫹ r
x
x2
rx
⫹
s
sx
rs
Value of c
B.
Length
x
Value of c
Chapter 4
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E.
Look for a pattern in the table for part D. Use this pattern to predict
the length and width of a rectangle with each of the following areas.
i) x 2 - 2x - 15
iii) x 2 - x - 30
ii) x 2 + 2x - 24
iv) x 2 - 8x + 7
F.
The expression x 2 + bx + c represents the area of a rectangle.
How can you factor this expression to predict the length and width
of the rectangle?
Reflecting
G.
Examine the length, width, and area of each rectangle in parts A
and D. Explain how the signs in the area expression can be used
to determine the signs in each dimension.
H.
Can all quadratic expressions of the form x 2 + bx + c be factored
as the product of two binomials? Explain.
APPLY the Math
EXAMPLE 1
Selecting an algebra tile strategy to factor a quadratic expression
Factor x 2 - 2x - 8.
Timo’s Solution
1
1
x
x
⫺1 ⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1 ⫺1
I arranged one x 2 tile, four ⫺x tiles, two x tiles, and
eight negative unit tiles in a rectangle to create an
area model.
x
x2
⫺x ⫺x ⫺x ⫺x
x
⫺1 ⫺1 ⫺1 ⫺1
I placed the x tiles and unit tiles so the length and
width were easier to see. The dimensions of the
rectangle are x ⫺ 4 and x ⫹ 2.
x 2 - 2x - 8 = (x - 4)(x + 2)
EXAMPLE 2
The sum 2 ⫹ (⫺4) ⫽ ⫺2 determines the number of
x tiles, and the product 2 ⫻ (⫺4) ⫽ ⫺8 determines
the number of unit tiles in the original expression.
Connecting the factors of a trinomial to its coefficients and constants
Factor x 2 + 12x + 35.
Chaniqua’s Solution
x 2 + 12x + 35
= (x ? )(x ? )
208
4.3 Factoring Quadratics: x 2 ⴙ bx ⴙ c
The two factors of the quadratic expression must
be binomials that start with x. I need two numbers
whose sum is 12 (the coefficient of x) and whose
product is 35 (the constant).
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4.3
I started with the product. Since 35 is positive, both
numbers must be either positive or negative.
= (x + ?)(x + ?)
Since the sum is positive, both numbers must be
positive.
= (x + 7)(x + 5)
The numbers are 7 and 5.
Check:
(x + 7)(x + 5) = x 2 + 5x + 7x + 35
= x 2 + 12x + 35
I checked by multiplying.
EXAMPLE 3
Reasoning to factor quadratic expressions
Factor each expression, if possible.
a) x 2 - x - 72
b) a 2 - 13a + 36
c) x 2 + x + 6
Ryan’s Solution
a) x 2 - x - 72
= (x - 9)(x + 8)
I needed two numbers whose sum is ⫺1 and
whose product is ⫺72.
The product is negative, so one of the numbers
must be negative.
Since the sum is negative, the negative number
must be farther from zero than the positive
number.
The numbers are ⫺9 and 8.
b) a 2 - 13a + 36
= (a - 9)(a - 4)
I needed two numbers whose sum is ⫺13 and
whose product is 36.
The product is positive, so both numbers must be
either positive or negative.
Since the sum is negative, both numbers must be
negative.
The numbers are ⫺9 and ⫺4.
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Chapter 4
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I needed two numbers whose sum is 1 and whose
product is 6.
c) x 2 + x + 6
There are no such numbers because the only
factors of 6 are 1 and 6, and 2 and 3. The sum of 1
and 6 is 7, and the sum of 2 and 3 is 5. Neither
sum is 1.
This cannot be factored.
EXAMPLE 4
Reasoning to factor a quadratic that has a common factor
Factor 3y 3 - 21y 2 - 24y.
Sook Lee’s Solution
3y 3 - 21y 2 - 24y
First, I divided out the greatest common factor. The
GCF is 3y, since all the terms are divisible by 3y.
= 3y(y 2 - 7y - 8)
To factor the trinomial, I needed two numbers
whose sum is ⫺7 and whose product is ⫺8. These
numbers are ⫺8 and 1.
= 3y(y - 8)(y + 1)
In Summary
Key Idea
• If a quadratic expression of the form x 2 ⫹ bx ⫹ c can be factored, it can
be factored into two binomials, (x ⫹ r) and (x ⫹ s), where r ⫹ s ⫽ b and
r ⫻ s ⫽ c, and r and s are integers.
Need to Know
• To factor x 2 ⫹ bx ⫹ c as (x ⫹ r)(x ⫹ s), you can use the signs
in the trinomial to determine the signs in the factors.
Trinomial
Factors
b and c are positive.
(x ⫹ r)(x ⫹ s)
b is negative, and c is positive.
(x ⫺ r)(x ⫺ s)
b and c are negative.
(x ⫺ r)(x ⫹ s), where r ⬎ s
b is positive, and c is negative.
(x ⫹ r)(x ⫺ s), where r ⬎ s
• It is easier to factor an algebraic expression if you first divide out
the greatest common factor.
210
4.3 Factoring Quadratics: x 2 ⴙ bx ⴙ c
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4.3
CHECK Your Understanding
1. a) Write the trinomial that is represented by these algebra tiles.
x2
x x x x x
1 1 1 1 1 1
b) Sketch what the tiles would look like if they were arranged
in a rectangle.
c) Use your sketch to determine the factors of the trinomial.
2. The tiles in each model represent an algebraic expression. Identify
the expression and its factors.
a)
x
x
⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1
x2
⫺x ⫺x ⫺x
b)
⫺x
⫺x
⫺x
1 1 1 1
1 1 1 1
1 1 1 1
x2
⫺x ⫺x ⫺x ⫺x
3. One factor is given, and one factor is missing. What is the missing
factor?
a) x 2 - 10x + 21 = (x - 7) (.)
b) x 2 + 4x - 32 = (x - 4)(.)
c) x 2 - 2x - 63 = (.)(x + 7)
d) x 2 + 14x + 45 = (.)(x + 9)
PRACTISING
4. Factor each expression.
a) x 2 + 2x + 1
b) x 2 - 2x + 1
c) x 2 - 2x - 3
d) x 2 + 6x + 9
e) x 2 - 4x + 4
f ) x 2 - 4x - 12
5. The tiles in each model represent a quadratic expression. Identify
the expression and its factors.
a)
NEL
b)
x2
x x x x x
x2
⫺x ⫺x
⫺x
⫺x
⫺1 ⫺1 ⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1 ⫺1 ⫺1
⫺x
⫺x
1 1
1 1
Chapter 4
211
6. One factor is given, and one factor is missing. What is the missing
factor?
a) x 2 + 11x + 24 = (x + 3)(.)
b) c 2 - 15c + 56 = (c - 7)(.)
c) a 2 - 11a - 60 = (a - 15)(.)
d) y 2 - 20y - 44 = (.)(y + 2)
e) b 2 + 2b - 48 = (.)(b + 8)
f ) z 2 - 19z + 90 = (.)(z - 10)
7. Factor each expression.
a) x 2 + 4x + 3
b) a 2 - 9a + 20
c) m2 - 8m + 16
d) n 2 + n - 6
e) x 2 + 6x - 16
f ) x 2 + 15x - 16
8. Factor.
a) x 2 - 10x + 16
b) y 2 + 6y - 40
c) a 2 - a - 56
d) w 2 - 5w - 14
e) m 2 - 12m + 32
f ) n 2 + n - 42
9. Factor.
a) 3x 2 + 24x + 45
b) 2y 2 - 2y - 60
c) 3v 2 + 9v + 6
d) 6n 2 + 24n - 30
e) x 3 + 5x 2 + 4x
f ) 7x 4 + 28x 3 - 147x 2
10. Write three different quadratic trinomials that have (x - 2)
K
as a factor.
11. Nathan factored x 2 - 15x + 44 as (x - 4)(x - 11). Martina
C
factored the expression another way and found different factors.
Identify the factors that Martina found, and explain why both
students are correct.
12. Factor.
a) a 2 + 8a + 15
b) 3x 2 - 21x - 54
c) z 2 - 16z + 55
d) x 2 + 5x - 50
e) x 3 - 3x 2 - 10x
f ) 2xy 2 - 26xy + 84x
13. Examine each quadratic relation below.
i) Express the relation in factored form.
ii) Determine the zeros.
iii) Determine the coordinates of the vertex.
iv) Sketch the graph of the relation.
a) y = x 2 + 2x - 8
b) y = x 2 - 2x - 24
c) y = x 2 - 8x + 15
d) y = - x 2 - 9x - 14
212
4.3 Factoring Quadratics: x 2 ⴙ bx ⴙ c
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4.3
14. A professional cliff diver’s height above the water can be modelled by
A
the equation h = - 5t 2 + 20, where h is the diver’s height in metres
and t is the elapsed time in seconds.
a) Draw a height versus time graph.
b) Determine the height of the cliff that the diver jumped from.
c) Determine when the diver will enter the water.
15. A baseball is thrown from the top of a building and falls to the ground
below. The height of the baseball above the ground is approximated
by the relation h = - 5t 2 + 10t + 40, where h is the height above
the ground in metres and t is the elapsed time in seconds. Determine
the maximum height that is reached by the ball.
16. Factor each expression.
a) m 2 + 4mn - 5n 2
b) x 2 + 12xy + 35y 2
c) a 2 + ab - 12b 2
d) c 2 - 12cd - 85d 2
e) r 2 + 13rs + 12s 2
f ) 18p 2 - 9pq + q 2
17. Paul says that if you can factor x 2 - bx - c, you can factor
T
x 2 + bx - c. Do you agree? Explain.
18. Create a mind map that shows the connections between x 2 + bx + c
and its factors.
Extending
19. Factor each expression.
a) x 4 + 6x 2 - 27
b) a 4 + 10a 2 + 9
c) -4m 4 + 16m 2n 2 + 20n 4
d) (a - b)2 - 15(a - b) + 26
20. Factor, and then simplify. Assume that the denominator is never zero.
x 2 - 6x + 8
x - 4
2
a - 3a - 28
b)
a + 4
a)
NEL
x 2 + x - 30
x - 5
2
2x - 24x + 64
d)
2x - 16
c)
Chapter 4
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
Q:
How do you determine the greatest common factor
of the terms in a polynomial?
A1:
Sometimes you can represent the terms with algebra tiles and arrange
the tiles into rectangles. The arrangement that has the greatest
possible width shows the greatest common factor.
• See Lesson 4.1,
Examples 1 to 4.
• Try Mid-Chapter Review
Questions 1 to 6.
EXAMPLE
Factor 4x 2 - 6x.
Solution
The greatest common factor of the terms in 4x 2 - 6x is 2x, since this
is the greatest width possible in a rectangular arrangement of tiles.
x2
x2
x x x
x2
x2
x x x
2x
x
x2
2x 3
x2
x2
x2
x x x x x x
4x 6
Once you divide out the common factor, the remaining terms represent
the other dimension of the rectangle.
4x 2 - 6x = 2x(2x - 3)
A2:
You can determine the greatest common factor of the coefficients and
the greatest common factor of the variables, and then multiply
the GCFs together. Sometimes you may need to group terms since
the GCF can be a monomial or a binomial.
EXAMPLE
Factor 5xa - 5xb + 2ya - 2yb.
Solution
When you group terms,
5xa - 5xb 2ya - 2yb 5x(a - b) + 2y(a - b),
and the GCF is (a - b).
Divide out the common factor.
5xa - 5xb + 2ya - 2yb (a - b)(5x + 2y)
214
Mid-Chapter Review
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Mid-Chapter Review
Q:
How can you factor a quadratic expression of the form
x 2 ⴙ bx ⴙ c?
A1:
You can form a rectangular area model using tiles. The length and
width of the rectangle are the factors.
Study
Aid
• See Lesson 4.3,
Examples 1 to 3.
• Try Mid-Chapter Review
Questions 7 to 13.
EXAMPLE
Factor x 2 + 11x + 18.
Solution
Algebra tiles can be arranged to form this rectangle.
x9
x2
x x x x x x x x x
x
x
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
x2
This rectangle has a width of x + 2 and a length of x + 9.
These are the factors of x 2 + 11x + 18.
x 2 + 11x + 18 = (x + 9)(x + 2)
A2:
You can look for values whose sum is b and whose product is c, and
then use these values to factor the expression.
EXAMPLE
Factor x 2 - 3x - 40.
Solution
Pairs of numbers whose product is 40 are
• 40 and 1
• 40 and 1
• 20 and 2
• 20 and 2
• 10 and 4
• 10 and 4
• 8 and 5
• 8 and 5
The only pair whose sum is 3 is 8 and 5.
x 2 - 3x - 40 = (x - 8)(x + 5)
Expanding verifies this result.
(x - 8)(x + 5)
= x 2 + 5x - 8x - 40
= x 2 - 3x - 40
NEL
Chapter 4
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PRACTICE Questions
Lesson 4.1
Lesson 4.3
1. State the GCF for each pair of terms.
a) 24 and 60
b) x 3 and x 2
c) 10y and 5y 2
d) - 8a 2b and - 12ab 2
e) c 4d 2 and c 3d
f ) 27m 4n 2 and 36m 2n 3
7. Each model represents a quadratic expression.
Identify the expression and its factors.
a)
b)
1 1 1
x
x
1 1
2. Determine the missing factor.
a) 7x - 28y = (.)(x - 4y)
b) 6x - 9y = (3)(.)
c) 24a 2 + 12b 2 = (.)(2a 2 + b 2)
d) x 2 - xy 3 = (x)(.)
e) - 15x 2 + 6y 2 = (.)(5x 2 - 2y 2)
f ) a 4b 3 - a 3b 2 = (a 3b 2)(.)
3. The tiles in each model represent an algebraic
expression. Identify the expression and
the greatest common factor of its terms.
a)
x
x
x
1 1
x
x
x
x
x2
x2
x
x
1
1
1
1
b)
x2
x2
x x x x x x
x x x x x x
4. Factor each expression.
a) 7z + 35
b) - 28x 2 + 4x 3
c) 5m 2 - 10mn + 5
d) x 2y 4 - xy 2 + x 3y
5. A parabola is defined by the equation
y = 5x 2 - 15x. Explain how you would
determine the coordinates of the vertex of the
parabola, without using a table of values or
graphing technology.
6. Factor each expression.
a) 3x(5y - 2) + 5(5y - 2)
b) 4a(b + 6) - 3(b + 6)
c) 6xt - 2xy - 3t + y
d) 4ab + 4ac - b 2 - bc
216
Mid-Chapter Review
x
x
x
1
1
1
x2
x x
1
1
1
x2
x x x
8. Determine the value of each symbol.
a) x 2 + 䉬x + 12 = (x + 3)(x + .)
b) x 2 + . x + 䉬 = (x + 3)(x + 3)
c) x 2 - 12x + . = (x - 䉬)(x - 䉬)
d) x 2 - 7x + 䉬 = (x - 3)(x - .)
9. Factor.
a) x 2 + 8x - 33
b) n 2 + 7n - 18
c) b 2 - 10b - 11
d) x 2 - 14x + 45
e) c 2 + 5c - 14
f ) y 2 - 17y + 72
10. Factor.
a) 3a 2 - 3a - 36 d) 4b 2 - 36b + 72
b) x 3 - 6x 2 - 16x e) - d 3 + d 2 + 30d
c) 2x 2 + 14x - 120 f ) xy 3 + 2xy 2 + xy
11. Deanna throws a rock from the top of a cliff
into the air. The height of the rock above the
base of the cliff is modelled by the equation
h = - 5t 2 + 10t + 75, where h is the height of
the rock in metres and t is the time in seconds.
a) How high is the cliff?
b) When does the rock reach its maximum
height?
c) What is the rock's maximum height?
12. When factoring a quadratic expression of the
form x 2 + bx + c, why does it make more
sense to consider the value of c before the value
of b? Explain.
13. Use the quadratic relation determined by
y = x 2 + 4x - 21.
a) Express the relation in factored form.
b) Determine the zeros and the vertex.
c) Sketch its graph.
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Factoring Quadratics:
ax2 + bx + c
YOU WILL NEED
GOAL
Factor quadratic expressions of the form ax2 + bx + c , where a ≠1.
LEARN ABOUT the Math
• algebra tiles
y
Kellie was asked to determine the x-intercepts of
y = 3x 2 + 11x + 6 algebraically. She created a graph
using graphing technology and estimated that the x-intercepts
are about x = - 0.6 and x = - 3.
x
Kellie knows that if she can write the equation in factored form,
she can use the factors to determine the x-intercepts. She is unsure
about how to proceed because the first term in the expression
has a coefficient of 3 and there is no common factor.
How can you factor 3x2 + 11x + 6 ?
?
EXAMPLE 1
Selecting a strategy to factor a trinomial,
where a Z 1
Factor 3x 2 + 11x + 6, and determine the x-intercepts
of y = 3x 2 + 11x + 6.
Ellen’s Solution: Selecting an algebra tile model
1
1
1
x
x
x
x
x
x
x
x
x
1
1
1
1
1
1
x
x2
x2
x2
x x
x
x
x
1
1
3x 2 + 11x + 6 = (3x + 2)(x + 3)
NEL
I used tiles to create a
rectangular area model
of the trinomial.
I placed the tiles along the
length and width to read
off the factors. The length
is 3x ⫹ 2, and the width
is x ⫹ 3.
Chapter 4
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The equation in factored form is
y = (3x + 2)(x + 3).
Let 3x + 2 = 0 or x + 3 = 0.
3x = - 2
x = -3
2
x = 3
The x-intercepts occur when
y ⫽ 0. This happens when
either factor is equal to 0.
2
The x-intercepts are - and ⫺3.
3
Neil’s Solution: Selecting an area diagram and
a systematic approach
qx
⫹ s
px
pqx2
psx
⫹
r
qrx
rs
I thought about the general
situation, where (px ⫹ r)
and (qx ⫹ s) represent the
unknown factors. I created
an area model and used it
to look for patterns
between the coefficients
in the factors and the
coefficients in the trinomial.
(px + r)(qx + s) = pqx 2 + psx + qrx + rs
= pqx 2 + (ps + qr)x + rs
Suppose that
3x 2 + 11x + 6 = (px + r)(qx + s).
I imagined writing two
factors for this product.
I had to figure out the
coefficients and the
constants in the factors.
(px + r)(qx + s) = pqx 2 + ( ps + qr)x + rs
= 3x 2 + 11x + 6
I matched the coefficients
and the constants.
p
q
r
s
ps ⫹ qr
3
1
3
2
9
1
3
2
3
9
1
3
3
2
11
冑
p = 1, q = 3, r = 3, and s = 2
3x 2 + 11x + 6 = (x + 3)(3x + 2)
218
4.4 Factoring Quadratics: ax 2 ⴙ bx ⴙ c
I needed values of p and
q that, when multiplied,
would give a product of 3.
I also needed values of r
and s that would give
a product of 6.
The middle coefficient is
11, so I tried different
combinations of ps ⫹ qr
to get 11.
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4.4
The equation in factored form is
y = (x + 3)(3x + 2).
Let x + 3 = 0 or 3x + 2 = 0.
x = -3
3x = - 2
2
x = 3
The x-intercepts occur
when y ⫽ 0. This happens
when either factor is equal
to zero.
2
The x-intercepts are ⫺3 and - .
3
Astrid’s Solution: Selecting a decomposition strategy
3x 2 ⫹ 11x ⫹ 6
⫽ (px ⫹ r)(qx ⫹ s)
I imagined writing two
factors for this product.
I had to figure out the
coefficients and the
constants in the factors.
(px ⫹ r)(qx ⫹ s)
I multiplied the binomials.
I noticed that I would get
the product of all four
missing values if I
multiplied the coefficient
of x2 (pq) and the
constant (rs).
⫽ pxqx ⫹ pxs ⫹ rqx ⫹ rs
⫽ pqx2 ⫹ (qr ⫹ ps)x ⫹ rs
ps and qr, the two values that are
added to get the coefficient of the
middle term, are both factors of pqrs.
3x 2 + 11x + 6
= 3x 2 + ?x + ?x + 6
3 ⫻ 6 ⫽ 18
The factors of 18 are 1, 2, 3, 6, 9, and 18.
11 ⫽ 9 ⫹ 2
If I added the product of
two of these values (ps) to
the product of the other
two (qr), I would get the
coefficient of x.
I needed to decompose
the 11 from 11x into two
parts. Each part had to be
a factor of 18, because
3 ⫻ 6 ⫽ 18.
3x 2 + 9x + 2x + 6
= 3x 2 + 9x + 2x + 6
= 3x(x + 3) + 2(x + 3)
I divided out the greatest
common factors from the
first two terms and then
from the last two terms.
= (x + 3)(3x + 2)
I factored out the binomial
common factor.
NEL
decompose
break a number or an expression
into the parts that make it up
Chapter 4
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The equation in factored form is
y ⫽ (x ⫹ 3)(3x ⫹ 2).
Let x ⫹ 3 ⫽ 0 or 3x ⫹ 2 ⫽ 0.
x ⫽ ⫺3
3x ⫽ ⫺2
2
x = 3
The x-intercepts occur when
y ⫽ 0. This happens when either
factor is equal to zero.
2
The x-intercepts are ⫺3 and - .
3
Reflecting
A.
Explain how Ellen’s algebra tile arrangement shows the factors
of the expression.
B.
How is Neil’s strategy similar to the strategy used to factor trinomials
of the form x 2 + bx + c? How is it different?
C.
How would Astrid’s decomposition change if she had been
factoring 3x 2 + 22x + 24 instead?
D.
Which factoring strategy do you prefer? Explain why.
APPLY the Math
EXAMPLE 2
Selecting a systematic strategy to factor a trinomial, where a Z 1
Factor 4x 2 - 8x - 5.
Katie’s Solution
4x 2 - 8x - 5 = (px + r)(qx + s)
= pqx 2 + (ps + qr)x + rs
pq ⫽ 4 and rs ⫽ ⫺5
p
q
r
s
1
4
1
⫺5
4
1
⫺5
1
2
2
pqx 2 + (ps + qr)x + rs = 4x 2 - 8x - 5
ps + qr = - 8
220
4.4 Factoring Quadratics: ax 2 ⴙ bx ⴙ c
I wrote the quadratic as the
product of two binomials with
unknown coefficients and
constants. Then I listed all the
possible pairs of values for pq
and rs.
I had to choose values that
would make ps ⫹ qr ⫽ ⫺8.
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4.4
(px + r)(qx + s) = (2x + 1)(2x - 5)
So, 4x 2 - 8x - 5 = (2x + 1)(2x - 5).
(2x + 1)(2x - 5) = 4x 2 - 10x + 2x - 5
= 4x 2 - 8x - 5
EXAMPLE 3
The values p ⫽ 2, q ⫽ 2, r ⫽ 1, and s ⫽ ⫺5
work because
• pq is (2)(2) ⫽ 4
• rs is (1)(⫺5) ⫽ ⫺5
• ps ⫹ qr is (2)(⫺5) ⫹ (2)(1) ⫽ – 8
I checked by multiplying.
Selecting a decomposition strategy to factor a trinomial
Factor 12x 2 - 25x + 12.
Braedon’s Solution
12x 2 - 25x + 12
= 12x 2 - 16x - 9x + 12
I looked for two numbers whose sum is ⫺25 and
whose product is (12)(12) ⫽ 144. I knew that
both numbers must be negative, since the sum
is negative and the product is positive. The
numbers are ⫺16 and ⫺9. I used these numbers
to decompose the middle term.
= 12x 2 - 16x - 9x + 12
= 4x(3x - 4) - 3(3x - 4)
= (3x - 4)(4x - 3)
I factored the first two terms and then the last
two terms. Then I divided out the common
factor of 3x ⫺ 4.
EXAMPLE 4
Selecting a guess-and-test strategy to factor a trinomial
Factor 7x 2 + 19x - 6.
Dylan’s Solution
7x 2 + 19x - 6
x
7x
?
7x2
?
?
⫺6
7x
⫺6
x
7x2
⫺6x
1
7x
⫺6
?
(7x - 6)(x + 1) = 7x 2 + x - 6
NEL
I thought of the product of the factors as
the dimensions of a rectangle with the area
7x 2 + 19x - 6.
The only factors of 7x 2 are 7x and x.
The factors of ⫺6 are ⫺6 and 1, ⫺2 and 3,
6 and ⫺1, and 2 and –3. I had to determine
which factors of 7x 2 and ⫺6 would add to 19x.
I used trial and error to determine the values
in place of the question marks. Then I checked
by multiplying.
wrong factors
Chapter 4
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7x
⫺2
x
7x2
⫺2x
3
21x
⫺6
(7x - 2)(x + 3) = 7x 2 + 19x - 6
Page 222
I repeated this process until I found
the combination that worked.
worked
7x 2 + 19x - 6 = (7x - 2)(x + 3)
In Summary
Key Idea
• If the quadratic expression ax2 ⫹ bx ⫹ c (where a Z 1) can be factored,
then the factors have the form (px ⫹ r)(qx ⫹ s), where pq ⫽ a, rs ⫽ c,
and ps ⫹ rq ⫽ b.
Need to Know
• If the quadratic expression ax2 ⫹ bx ⫹ c (where a Z 1) can be
factored, then the factors can be found by
• forming a rectangle using algebra tiles
• using the algebraic model (px ⫹ r)(qx ⫹ s) ⫽ pqx2 ⫹ (ps ⫹ qr)x ⫹ rs
systematically
• using decomposition
• using guess and test
• A trinomial of the form ax2 ⫹ bx ⫹ c (where a Z 1) can be factored
if there are two integers whose product is ac and whose sum is b.
CHECK Your Understanding
1. a) Write the trinomial that is represented by the algebra tiles at the left.
x2
x x x
x2
1
b) Sketch what the tiles would look like if they were arranged
in a rectangle.
c) Use your sketch to determine the factors of the trinomial.
2. Each of the following four diagrams represents a trinomial. Identify
the trinomial and its factors.
a)
1
x
x
222
x
x
x2
x2
4.4 Factoring Quadratics: ax 2 ⴙ bx ⴙ c
b)
⫺x
⫺x
⫺x
1
x2
x2
x2
⫺x
x2
x2
x2
⫺x
1
x
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4.4
c)
?
?
?
8x2
6x
20x
?
d)
?
?
?
15x2
⫺5x
?
18x
⫺6
15
3. Determine the missing factor.
a) 2c 2 + 7c - 4 = (c + 4)(.)
b) 4z 2 - 9z - 9 = (.)(z - 3)
c) 6y 2 - y - 1 = (3y + 1)(.)
d) 6p 2 + 7p - 3 = (.)(2p + 3)
PRACTISING
4. Determine the value of each symbol.
a) 5x 2 + 䉬x + 3 = (x + 3)(5x + .)
b) 2x 2 - .x - 䉬 = (2x + 3)(x - 2)
c) 12x 2 - 7x + 䊏 = (3x - 䉬)(4x - 䉬)
d) 14x 2 - 29x + 䉬 = (2x - 3)(7x - .)
5. Factor each expression.
a) 2x 2 + x - 6
b) 3n 2 - 11n - 4
c) 10a 2 + 3a - 1
d) 4x 2 - 16x + 15
e) 2c 2 + 5c - 12
f ) 6x 2 + 5x + 1
6. Factor.
a) 6x 2 - 13x + 6
b) 10m 2 + m - 3
c) 2a 2 - 11a + 12
d) 4x 2 - 20x + 25
e) 5d 2 + 8 - 14d
f ) 6n 2 - 20 + 26n
7. Factor.
a) 15x 2 + 4x - 4
b) 18m 2 - 3m - 10
c) 16a 2 - 50a + 36
d) 35x 2 - 27x - 18
e) 63n 2 + 126n + 48
f ) 24d 2 + 35 - 62d
8. Write three different quadratic trinomials of the form ax 2 + bx + c,
where a Z 1, that have (3x - 4) as a factor.
9. The area of a rectangle is given by each of the following trinomials.
K
NEL
Determine expressions for the length and width of the rectangle.
a) A = 6x 2 + 17x - 3
b) A = 8x 2 - 26x + 15
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10. Identify possible integers, k, that allow each quadratic trinomial
T
to be factored.
a) kx 2 + 5x + 2
b) 9x 2 + kx - 5
c) 12x 2 - 20x + k
11. Factor each expression.
a) 6x 2 + 34x - 12
b) 18v 2 + 33v - 30
c) 48c 2 - 160c + 100
d) 5b 3 - 17b 2 + 6b
e) -6x - 51xy + 27xy 2
f ) -7a 2 - 29a + 30
12. Determine whether each polynomial has (k + 5) as one of its factors.
a) k 2 + 9k - 52
b) 4k 3 + 32k 2 + 60k
c) 6k 2 + 23k + 7
d) 10 + 19k - 15k 2
e) 7k 2 + 29k - 30
f ) 10k 2 + 65k + 75
13. Examine each quadratic relation below.
i) Express the relation in factored form.
ii) Determine the zeros.
iii) Determine the coordinates of the vertex.
iv) Sketch the graph of the relation.
a) y = 2x 2 - 9x + 4
b) y = - 2x 2 + 7x + 15
14. A computer software company models the profit on its latest video
A
game using the relation P = - 4x 2 + 20x - 9, where x is the number
of games produced in hundred thousands and P is the profit
in millions of dollars.
a) What are the break-even points for the company?
b) What is the maximum profit that the company can earn?
c) How many games must the company produce to earn
the maximum profit?
15. Factor each expression.
a) 8x 2 - 13xy + 5y 2
b) 5a 2 - 17ab + 6b 2
c) -12s 2 - sr + 35r 2
d) 16c 4 + 64c 2 + 39
e) 14v 6 - 39v 3 + 27
f ) c 3d 3 + 2c 2d 2 - 8cd
16. Create a flow chart that would help you decide which strategy
C
you should use to factor a given polynomial.
Extending
17. Factor.
a) 6(a + b)2 + 11(a + b) + 3
b) 5(x - y)2 - 7(x - y) - 6
c) 8(x + 1)2 - 14(x + 1) + 3
d) 12(a - 2)4 + 52(a - 2)2 - 40
18. Can a quadratic expression of the form ax 2 + bx + c always be
factored if b 2 - 4ac is a perfect square? Explain.
224
4.4 Factoring Quadratics: ax 2 ⴙ bx ⴙ c
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Factoring Quadratics:
Special Cases
YOU WILL NEED
GOAL
Factor perfect-square trinomials and differences of squares.
• algebra tiles
LEARN ABOUT the Math
Nadia claims that the equation y = 4x 2 + 12x + 9 will always generate
a value that is a perfect square, if x represents any natural number.
?
How can you show that Nadia’s claim is correct?
EXAMPLE 1
Connecting an expression to its factors
Show that Nadia’s claim is correct.
Parma’s Solution: Selecting an algebra tile model
x
y
Perfect Square?
1
25
Yes: 25 5 5
3
81
Yes: 81 9 9
9
441
Yes: 441 21 21
1
1
1
x
x
x
x
x
x
1 1 1
1 1 1
1 1 1
x
x2
x2
x x x
x
x2
x2
x x x
x
x
1 1 1
First I substituted some numbers
for x into the equation. Each
time, the result was a perfect
square. Nadia’s claim seems
to be correct.
I decided to see how algebra
tiles could be arranged. The only
arrangement of tiles that
seemed to work was a square.
I lined up tiles along two edges
to determine the side length
of the square.
Each side is (2x + 3) long.
4x 2 + 12x + 9 = (2x + 3)2
NEL
Chapter 4
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y = 4x 2 + 12x + 9
is the same as y = (2x + 3)2.
The equation y = 4x 2 + 12x + 9
will always result in a perfect square.
The equation factors and the
binomial factors are identical.
Any number that is substituted
for x gets squared. This ensures
that the result will always be
a perfect square.
Jarrod’s Solution: Reasoning logically to factor
y = 4x 2 + 12x + 9
4 = 22 and 9 = 32
I noticed that both 9 and 4 in
the equation are perfect squares.
Does 4x 2 + 12x + 9 = (2x + 3)2?
I tested to see if the trinomial is
a perfect square, with identical
factors, by expanding
(2x 3)(2x 3).
(2x + 3)2 = (2x + 3)(2x + 3)
= 4x 2 + 6x + 6x + 9
= 4x 2 + 12x + 9 worked
4x 2 + 12x + 9 = (2x + 3)2
y = 4x 2 + 12x + 9
is the same as y = (2x + 3)2.
The equation y = 4x 2 + 12x + 9
will always generate a perfect square.
This worked, since I got
the correct coefficient of x.
The equation has identical
factors. Any number that is
substituted for x gets squared.
This ensures that the result will
always be a perfect square.
Reflecting
226
A.
Why is the name “perfect-square trinomial” suitable for a polynomial
like 4x 2 + 12x + 9?
B.
Why did Jarrod use the square root of 4 and the square root of 9
as values in the factors?
C.
Nigel said that a quadratic expression cannot be a perfect square unless
the coefficient of x is even. How do Parma’s and Jarrod’s solutions
show that Nigel is correct?
4.5 Factoring Quadratics: Special Cases
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4.5
APPLY the Math
EXAMPLE 2
Connecting decomposition with factoring
a perfect square
Factor 25x 2 - 40x + 16.
Andy’s Solution
For the expression
25x 2 - 40x + 16,
- 40 = ( -20) + (- 20)
and ( - 20)( - 20) = 400
I decomposed 40, the
coefficient of x, as the sum of
two numbers whose product
is 25 16 400.
25x 2 - 20x - 20x + 16
I wrote the x term in
decomposed form.
25x 2 - 20x - 20x + 16
= 5x(5x - 4) - 4(5x - 4)
I divided out the GCF 5x from the
first two terms. I divided out the
GCF 4 from the last two terms.
= (5x - 4)(5x - 4)
= (5x - 4)2
Then I divided out the binomial
common factor.
EXAMPLE 3
Representing a difference of squares
as a square tile model
Factor x 2 - 1.
Lori’s Solution
1
x
1
x
x2
x
x
1
x 2 - 1 = (x + 1)(x - 1)
When using tiles to factor, the
unit tiles are always diagonal
from the x 2 tiles. I arranged my
tiles this way, but the rectangle
was not complete. I added one
vertical x tile and one horizontal
x tile to make a square. This is
the same as adding zero to the
expression, since x (x) 0.
The dimensions of the square
are (x 1) and (x 1).
NEL
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EXAMPLE 4
Page 228
Reasoning logically to factor a difference
of squares
Factor each expression.
a) x 2 - 64
b) 81x 4 - 25y 2
Natalie’s Solution
a) x 2 - 64, where
2x2 = x and 264 = 8
x 2 - 64 = (x + 8)(x - 8)
(x + 8)(x - 8)
= x 2 - 8x + 8x - 64
= x 2 - 64
b)
81x 4 - 25y 2, where
281x 4 = 9x 2 and
225y 2 = 5y
81x4 - 25y2 = (9x2 + 5y)(9x2 - 5y)
(9x 2 + 5y)(9x 2 - 5y)
= 81x 4 - 45x 2y + 45x 2y - 25y 2
= 81x 4 - 25y 2
228
4.5 Factoring Quadratics: Special Cases
This is a binomial. Both terms
are perfect squares, and they
are separated by a subtraction
sign. There is no x term.
I know, from expanding, that
this type of expression has
factors that are binomials. The
terms of the binomials contain
the square roots of the terms
of the original expression. One
binomial contains the sum of
the square roots, and the
other binomial contains the
difference.
I checked by multiplying.
When expanded, the x term
equals zero.
This is a difference of
squares. Both terms are
perfect squares, and they
are separated by a
subtraction sign.
Both factors are binomials. The
terms of the binomials contain
the square roots of the terms
of the original expression. One
binomial contains the sum of
the square roots. The other
binomial contains the
difference.
I checked by multiplying.
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4.5
In Summary
Key Ideas
• A polynomial of the form a2 2ab b2 or a2 2ab b2
is a perfect-square trinomial:
• a2 2ab b2 can be factored as (a b)2.
• a2 2ab b2 can be factored as (a b)2.
• A polynomial of the form a2 b2 is a difference of squares and
can be factored as (a b)(a b).
Need to Know
• A perfect-square trinomial and a difference of squares can be factored
by
• forming a square using algebra tiles
• using decomposition
• using logical reasoning
CHECK Your Understanding
1. a) Write the quadratic expression that is represented
by these algebra tiles.
x2
x
x
x
1 1 1
1 1 1
1 1 1
x
x
x
b) Sketch what the tiles would look like if they were arranged
in a rectangle.
c) Use your sketch to determine the factors of the trinomial.
2. Each model represents a quadratic expression. Identify the polynomial
and its factors.
a)
NEL
x
x
x
x
x
x
1 1
1 1
x2
x2
x2
x x
x2
x2
x2
x x
x2
x2
x2
b)
36x2
30x
30x
25
x x
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3. Determine the missing factor.
a) x 2 - 100 = (x + 10)(.)
b) n 2 + 10n + 25 = (.)(n + 5)
c) 81a 2 - 16 = (.)(9a - 4)
d) 20x 2 - 5 = (.)(2x - 1)(2x + 1)
e) 25m 2 - 70m + 49 = (.)2
f ) 18x 2 - 48x + 32 = 2(.)2
PRACTISING
4. Determine the value of each symbol.
a) 4x 2 + 䉬x + 25 = (2x + .)2
b) 25x 2 - 䉬 = (.x + 3)(.x - 3)
c) 16x 2 - 䉬x + 81 = (4x - .)(4x - .)
d) 䉬x 2 - 64 = (3x - .)(3x + .)
5. Factor each expression.
a) x 2 - 25
b) y 2 - 81
c) a 2 - 36
d) 4c 2 - 49
e) 9x 2 - 4
f ) 25d 2 - 144
6. Factor.
a) x 2 + 10x + 25 c) m 2 - 4m + 4
e) 16p 2 + 72p + 81
2
2
b) b + 8b + 16 d) 4c - 44c + 121 f ) 25z 2 - 30z + 9
7. Factor.
K
a) 49a 2 + 56a + 16
b) 4x 2 - 25
c) - 50x 2 - 40x - 8
d) 4a 2 - 256
e) 225 - 16x 2
f ) (x + 1)2 + 2(x + 1) + 1
8. You can use the pattern for the difference of squares to help you do
A
mental calculations. Show how you can do this for each expression.
a) 642 - 602
b) 182 - 122
9. Explain how you know that 8x 2 - 18x + 9 cannot be factored as
C
a) a perfect square
b) a difference of squares
10. Factor each expression.
a) x 4 - 12x 2 + 36
b) a 4 - 16
c) 49x 2 - 100
d) 12x 2 - 60x + 75
e) x 4 - 24x 2 + 144
f ) 289x 6 - 81
11. Factor.
a) x 2 – 16xy + 64y 2
b) 36x 2 – 25y 2
c) 16x 2 – 72xy + 81y 2
d) 1 - 9a 2b 4
e) –18x 2 + 24xy - 8y 2
f ) 50x 3 - 8xy 2
12. Factor each expression. Explain the strategy you used.
T
230
a) x 2 - c 2 - 8x + 16
4.5 Factoring Quadratics: Special Cases
b) 4c 2 - a 2 - 6ab - 9b 2
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4.5
13. Examine each quadratic relation below.
i) Express the relation in factored form.
ii) Determine the zeros.
iii) Determine the coordinates of the vertex.
iv) Sketch the graph of the relation.
a) y = - x 2 + 16x - 64
b) y = 4x 2 - 1
14. Determine a simplified expression for the area of each shaded region.
a)
b)
(x 2)
(4x 5)
(x 1)
(3x 2)
15. Copy and complete the following charts to show what you know
about each type of polynomial.
a) Definition:
Characteristics:
Examples:
Perfect
Square
Nonexamples:
b) Definition:
Examples:
Characteristics:
Difference
of Squares
Nonexamples:
Extending
16. a) Multiply (a + b)(a 2 - ab + b 2).
b) Compare your product for part a) with the factors of the original
expression. Identify the pattern you see.
c) Use the pattern you identified for part b) to factor each
expression.
i) x 3 + 8
iii) 8x 3 + 1
3
ii) x + 27
iv) 27x 3 + 8
17. a) Multiply (a - b)(a 2 + ab + b 2).
b) Compare your product for part a) with the factors of the original
expression. Identify the pattern you see.
c) Use the pattern you identified for part b) to factor each
expression.
i) x 3 - 27
iii) 8x 3 - 125
3
ii) x - 64
iv) 64x 3 - 27
NEL
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Curious Math
Creating Composite Numbers
Jamie claims that if you multiply any natural number greater than 1 by itself
four times and then add 4, the result will always be a composite number.
Savita made a chart and tested some numbers to see if Jamie’s claim could
be true.
Starting
Number
Calculation
Result
Prime or Composite?
2
22224
20
composite: 20 4 5
3
33334
85
composite: 85 5 17
5
55554
629
composite: 629 17 37
7
77774
2405
composite: 2405 5 481
1. Choose another starting number, and test this number to see if Jamie’s
claim holds.
2. Based on your observations, do you think Jamie’s calculation will
always result in a composite number? Explain.
3. Use n to represent any natural number, where n 1. Create an
algebraic expression that represents any result of Jamie’s calculation.
4. Add the terms 4n2 and -4n2 to the expression you created in step 3.
5. Factor the expression you created for step 4.
6. Explain why the factors you found for step 5 prove that Jamie’s claim
is true.
232
Curious Math
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Reasoning about Factoring
Polynomials
GOAL
Use reasoning to factor a variety of polynomials.
LEARN ABOUT the Math
The trinomial 8x 3 - 6x 2 - 5x represents the volume
of a rectangular prism.
?
?
What algebraic expressions represent the dimensions
of this prism?
?
?
EXAMPLE 1
Solving a problem using a factoring strategy
Determine the dimensions of the rectangular prism.
Rob’s Solution
V = l * w * h
The volume of a rectangular prism is calculated by multiplying
together its length, width, and height.
V = 8x 3 - 6x 2 - 5x
= x(8x 2 - 6x - 5)
I tried to factor the expression. Each term contained the common
factor of x, so I divided this out. The other factor was a trinomial,
where a Z 1.
= x(8x 2 - 10x + 4x - 5)
= x(8x 2 - 10x + 4x - 5)
= x[2x(4x - 5) + 1(4x - 5)]
= x(4x - 5)(2x + 1)
To factor the trinomial, I used decomposition. I looked for two
numbers whose sum is ⫺6 and whose product is (8)(⫺5) ⫽ ⫺40.
I knew that one number must be negative and the other number
must be positive, since the sum and the product are both negative.
The numbers are ⫺10 and 4. I used these numbers to decompose
the middle term.
I factored the first two terms of the expression in brackets and then
the last two terms. I divided out the common factor of 4x ⫺ 5.
= x(4x - 5)(2x + 1)
Possible dimensions of the
rectangular prism are length = 4x – 5,
width = 2x + 1, and height = x.
NEL
I can’t be sure which dimension is which, but I know
that the volume expression is correct when these terms
are multiplied together.
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Reflecting
A.
Why did Rob decide to use a factoring strategy?
B.
Why did he start by dividing out the common factor
from the expression?
C.
Why can he not be sure which factors correspond to which
dimensions of the prism?
APPLY the Math
EXAMPLE 2
Reasoning to factor polynomials
Factor each expression.
a) x 2 + x - 132
b) 16x 2 - 88x + 121
c) -18x 4 + 32x 2
Sunny’s Solution
a) x 2 + x - 132
= (x + 12)(x - 11)
This expression has three terms. It’s a trinomial,
and there are no common factors.
The factors must be two binomials that both start
with x. To determine the factors, I need two
numbers whose product is ⫺132 and whose sum
is 1. The numbers are 12 and ⫺11.
b) 16x 2 - 88x + 121
= (4x - 11)(4x - 11)
= (4x - 11)2
This expression has three terms. It’s a trinomial,
and there are no common factors.
The first and last terms are perfect squares. When
I doubled the product of their square roots, I got
the middle term.
This is a perfect-square trinomial.
c) - 18x 4 + 32x 2
= - 2x 2(9x 2 - 16)
= - 2x 2(3x - 4)(3x + 4)
234
4.6 Reasoning about Factoring Polynomials
This expression has two terms and a greatest
common factor of ⫺2x2. I divided out the GCF.
The factor 9x2 ⫺ 16 is a difference of squares.
The factors of 9x2 ⫺ 16 are binomials that contain
the square roots of each term.
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4.6
EXAMPLE 3
Selecting a grouping strategy to factor
Factor x 5y + x 2y 3 - x 3y 3 - y 5.
Monique’s Solution
x 5y + x 2y 3 - x 3y 3 - y 5
= y(x 5 + x 2y 2 - x 3y 2 - y 4)
= y(x 5 + x 2y 2 - x 3y 2 - y 4)
= y[x 2(x 3 + y 2) - y 2(x 3 + y 2)]
= y[(x 3 + y 2)(x 2 - y 2)]
= y(x 3 + y 2)(x - y)(x + y)
This expression has four terms, so I can’t use
strategies that work with trinomials. I divided out
the greatest common factor of y.
I grouped the first two terms and the last two
terms in the second factor, because I saw that the
first grouping had a common factor of x2 and the
second grouping had a common factor of y2.
I divided out the common factor from each pair.
Then I factored out x3 ⫹ y2, since it’s a common
factor.
Finally, I factored x2 ⫺ y2 using the pattern
for the difference of squares.
In Summary
Key Idea
• The strategy that you use to factor an algebraic expression depends on
the number of terms and the type of terms in the expression.
Need to Know
• You can use the following checklist to decide how to factor
an algebraic expression:
• Divide out all the common monomial factors.
• If the expression has two terms, check for a difference
of squares: a2 ⫺ b2 ⫽ (a ⫹ b)(a ⫺ b)
• If the expression has three terms, check for a perfect square:
a2 ⫹ 2ab ⫹ b2 ⫽ (a ⫹ b)2 or a2 ⫺ 2ab ⫹ b2 ⫽ (a ⫺ b)2
• If the expression has three terms and is of the form
x2 ⫹ bx ⫹ c, look for factors of the form (x ⫹ r)(x ⫹ s),
where c ⫽ r ⫻ s and b ⫽ r ⫹ s.
• If the expression has three terms and is of the form
ax2 ⫹ bx ⫹ c, a ≠ 1, look for factors of the form
(px ⫹ r)(qx ⫹ s), where c ⫽ r ⫻ s, a ⫽ p ⫻ q, and b ⫽ ps ⫹ qr.
• If the expression has four or more terms, try a grouping strategy.
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CHECK Your Understanding
1. Identify the type of algebraic expression and the factoring strategies
you would use to factor the expression.
a) 6xy + 12x 2y 2 - 4x 3y 3
d) 49y 2 - 9
2
b) 20x + 11x - 3
e) 3x 2 - 3x - 90
c) 3x 2 + 3xa - 2x - 2a
f ) x 2 - 13x + 42
2. Factor each expression in question 1.
PRACTISING
3. a) Create a factorable algebraic expression for each situation.
i) a perfect-square trinomial
ii) a trinomial of the form ax 2 + bx + c, where a = 1
iii) a difference of squares
iv) a trinomial of the form ax 2 + bx + c, where a Z 1
v) an expression that contains a monomial common factor
vi) an expression that contains a binomial common factor
b) Exchange the expressions you created with a partner. Factor
your partner’s expressions, and discuss the solutions.
4. Determine the value of each symbol.
a) -10a 3 + 15a 2 = - 5a(䉬a 2 - . a)
b) x 2 - 䉬x - 63 = (x + 7)(x - .)
c) 25x 2 - 䉬 = (. x - 7)(.x + 䊉)
d) 6x 2 + 䊉 x - 10 = (2x + .)(䉬x - 2)
5. Recall the question about working backwards on the opening page of
this chapter. Work backwards to determine the value of each symbol:
(䉱x + 䊉)(. x + 䉬) = 3x 2 + 11x + 10
6. Factor each expression.
K
a) 16x 2 - 25
b) -6b 2a - 9b 3 + 15b 2
c) c 2 - 12c + 35
d) 49d 2 + 14d + 1
e) 12x 2 + 4x - 21
f ) 2wz + 6w - 5z - 15
7. Factor.
a) 10x 2 + 3x - 1
b) 144a 4 - 121
c) 24ac - 8c + 21a - 7
d) x 3 - 11x 2 + 18x
e) 18x 2 + 60x + 50
f ) x 2y - 4y
8. Factor.
a) 2s 2 + 4s - 6
b) 14 - 5w - w 2
c) z 4 - 13z 2 + 36
236
4.6 Reasoning about Factoring Polynomials
d) 8s 2 - 50r 2
e) 36 - 84g + 49g 2
f ) x 2 + 10x + 25 - 16y 2
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4.6
9. Explain why each expression is not factored fully.
C
a) x 4 - 1 = (x 2 - 1)(x 2 + 1)
b) x 2y - 9xy + 20y = y(x 2 - 9x + 20)
c) 15x 2 + 6xy - 5x - 2y = 3x(5x + 2y) - (5x + 2y)
d) 48a 4c 3 - 3b 4c 3 = 3c 3(16a 4 - b 4)
10. Factor.
a) xy - ty + xs - ts
b) - 25x 2 + 16y 2
c) a 4 - 13a 2 + 36
d) x 2 - y 2 - 2y - 1
e) 2(a + b)2 + 5(a + b) + 3
f ) 6x 3 - 63x - 13x 2
11. The area of a rectangle is given by the relation A = 8x 2 + 18x + 7.
a) Determine expressions for the possible dimensions
of this rectangle.
b) Determine the dimensions and area of this rectangle if x = 3 cm.
12. The area of a circle is given by the relation A = px 2 + 10px + 25p.
A
a) Determine an expression for the radius of this circle.
b) Determine the radius and area of this circle if x = 10 cm.
13. The volume of a rectangular prism is given by the relation
V = 2x 3 + 14x 2 + 24x.
a) Determine expressions for the possible dimensions of this prism.
b) Determine the dimensions and volume of this prism if x = 5 cm.
14. Decide whether each polynomial has (x + y) as one of its factors.
T
Justify your decision.
a) xy + 3x 2y - 4xy + 6x 2y
b) x 5 - x 3y 2 + x 3 - xy 2
c) x 2y + 6y - 9xy + x 2y + xy
d) x 3 + 5x 2 + 6x + x 2y + 5xy + 6y
15. Create a graphic organizer that would help you decide which strategy
you should use to factor a given algebraic expression.
Extending
16. Factor each expression, if possible.
x2
1
9
4
b) 100 - (a - 5)2
c) 4x 2 + y 2
a)
9b 2
25a 2
64
49
2
e) (x + 3) - (y - 3)2
f ) 4(c - 5)4 + 12(c - 5)2 + 9
d)
x
17. A square with sides that measure x units is drawn. Then a square with
sides that measure y units is removed. Use the diagram to explain why
x 2 - y 2 = (x + y)(x - y).
y
x
y
NEL
Chapter 4
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Chapter Review
FREQUENTLY ASKED Questions
Aid
• See Lesson 4.4,
Q:
Examples 1 to 4.
• Try Chapter Review
Questions 8 to 11.
How can you factor a quadratic expression of the form
ax2 ⴙ bx ⴙ c, where a ⴝ 1?
A:
Some polynomials of this form can be factored, but others cannot.
Try to factor the expression using one of the methods below. If none
of these methods work, the expression cannot be factored.
x
x
x
x
x
x
x
x
x
x
x
x
1
1
1
1
x2
x2
x2
x
EXAMPLE
Factor 3x 2 + 11x - 4.
Solution
Method 1
1
1
1
1
x
x
x
x
x
x
x
x
x
x
x
x
1
1
1
1
x
x2
x2
x2
x
x
x
x
1
qx
The algebra tiles show that 3x 2 + 11x - 4 = (3x - 1)(x + 4).
You can check by multiplying.
(3x - 1)(x + 4) = 3x 2 + 12x - x - 4
= 3x 2 + 11x - 4
px
r
3x2
?
?
s
Arrange algebra tiles to form a rectangle. The rectangle for
3x 2 + 11x - 4 was not complete, so +x and -x were added
to complete it. The length and width are the factors.
3x
Method 2
Use guess and test with an area diagram.
If there are factors for 3x 2 + 11x - 4, you know that they are of the
form ( px + r)(qx - s) since the constant is negative. You also know that
p q 3, so the values of p and q can be 3 and 1, or 3 and 1. As well,
4
you know that r s 4, so the values of r and s can be 2 and 2,
4 and 1, or 1 and 4.
1
Try different combinations of p, q, r, and s until you find the combination
that gives ps + qr = 11.
The area diagram shows that 3x 2 + 11x - 4 = (3x - 1)(x + 4).
x
3x2
x
4
12x
4
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Chapter Review
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Chapter Review
Method 3
Use decomposition. Decompose 11 as the sum of two numbers
that multiply to 12 since 4 3 12. Use 1 and 12.
3x 2 + 11x - 4
= 3x 2 - 1x + 12x - 4
= x(3x - 1) + 4(3x - 1)
= (3x - 1)(x + 4)
Q:
How can you recognize a perfect-square trinomial or
a difference of squares, and how do you factor it?
A:
An expression is a perfect-square trinomial when the coefficient of x 2
is a perfect square, the constant is a perfect square, and the middle
coefficient is twice the product of the two square roots. For example,
25x 2 + 60x + 36 is a perfect square whose factors are (5x + 6)
and (5x + 6).
Study
Aid
• See Lesson 4.5,
Examples 1 to 4.
• Try Chapter Review
Questions 12 to14.
An expression is a difference of squares when it consists of two perfect
square terms and one of these terms is subtracted from the other.
For example, 100x 2 - 81 is a difference of squares whose factors
are (10x - 9) and (10x + 9).
Q:
How do you decide which strategy you should use to factor
an algebraic expression?
A:
The strategy you use depends on the type of expression you are given
and the number of terms it contains. You can follow this checklist
to help you decide which strategy to use:
• Divide out all the common monomial factors.
• If the expression has two terms, check for a difference
of squares: a 2 - b 2 = (a + b)(a - b)
• If the expression has three terms, check for a perfect square:
a 2 + 2ab + b 2 = (a + b)2 or a 2 - 2ab + b 2 = (a - b)2
2 + bx + c,
• If the expression has three terms and is of the form x
look for factors of the form (x + r)(x + s), where c = r * s and
b = r + s.
2 + bx + c,
• If the expression has three terms and is of the form ax
a 1, look for factors of the form ( px + r)(qx + s), where
c = r * s , a = p * q, and b = ps + qr.
• If the expression has four or more terms, try a grouping strategy.
NEL
Study
Aid
• See Lesson 4.6, Examples
1 to 3.
• Try Chapter Review
Questions 15 to 19.
Chapter 4
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PRACTICE Questions
7. Examine the relation y = x 2 + 7x + 12.
Lesson 4.1
1. Each model represents an algebraic expression.
Identify the expression and its factors.
a)
x
x
⫺1 ⫺1 ⫺1
x
x
x
x
⫺1 ⫺1 ⫺1
⫺1 ⫺1 ⫺1
a) Write the relation in factored form.
b) Determine the coordinates of the x-intercepts.
c) Determine the coordinates of the vertex.
d) State the minimum value of the relation and
where the minimum value occurs.
Lesson 4.4
b)
x2
8. Identify each expression that is modelled below,
x x x x
and state its factors.
a)
x2
x x x x
x2
10x
8x
80
2. Factor each expression.
a) 20x 2 - 4x
b) 3n 2 - 6n + 15
c) - 2x 3 + 6x 2 + 4x
d) 6a(3 - 7a) - 5(3 - 7a)
3. The area of a rectangle is given by the relation
A = 16x 2 - 24.
a) Determine possible dimensions
of this rectangle.
b) Is there more than one possibility? Explain.
b)
10x2
⫺4x
⫺35x
14
4. a) Write three polynomials whose terms have
a greatest common factor of 4x 3y.
b) Factor each polynomial you wrote for part a).
Lesson 4.2
5. Identify each expression that is modelled below,
and state its factors.
a)
x
⫺1
x
x
x
x2
⫺1
⫺1
⫺1
b)
x
x
x
x
x
x
⫺1
⫺1
⫺1
x2
x2
⫺x
⫺x
6. Factor each expression.
240
Chapter Review
each trinomial.
a) 15x 2 - 4x - 4
b) 20x 2 + 3x - 2
c) 7a 2 + 6a - 16
d) 20y 2 - 17y - 10
10. Factor each expression.
Lesson 4.3
a) x 2 + 16x + 63
b) x 2 - 7x - 60
9. Explain the strategy you would use to factor
c) x 2 + 6x - 27
d) 5x 2 - 5x - 100
a) 7x 2 - 19x - 6
b) 4a 2 + 23a + 15
c) 12x 2 - 16x + 5
d) 6n 2 - 11ny - 10y 2
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Chapter Review
11. Erica and Asif sell newly designed digital watches.
14. The polynomial x 2 - 25 can be factored. Can
The profit on the watches they sell is determined
by the relation P = - 2n 2 + 120n - 1000,
where n is the number of watches sold and
P is the profit in dollars.
the polynomial x 2 + 25 be factored? Explain.
Lesson 4.6
15. How is expanding an algebraic expression
related to factoring an algebraic expression?
Use an example in your explanation.
16. Factor each expression.
a) What are the break-even points for Erica
and Asif?
b) What is the maximum profit that Erica and
Asif can earn?
Lesson 4.5
12. Identify each expression that is modelled below,
and state its factors.
a)
x
x
x
x
x
x
1 1 1
1 1 1
1 1 1
x2
x2
x x x
x2
x2
x x x
a) 7x 2 - 26x - 8
b) 64a 6 - 25
c) 18ac - 12a - 15c + 10
d) 4x 2y - 44xy + 72y
e) 20x 2 + 61x + 45
f ) z 4 - 13z 2 + 40
17. Factor.
a) 2s 2 + 3s - 5
b) 15 - 2w - w 2
c) z 4 - 4z 2 - 32
d) 16s 2 - 121r 2
e) 9 - 30g + 25g 2
f ) x 2 + 16x + 64 - 25y 2
18. A packaging company creates different-sized
cardboard boxes. The volume of a box is given
by V = 18x 3 - 2x + 45x 2 - 5.
b)
64x2
⫺24x
24x
⫺9
13. Factor each expression.
a) 144x 2 - 25
b) 36a 2 + 12a + 1
c) 18x 5 - 512xy 2
d) 4(x - 2)2 - 20(x - 2) + 25
e) (x + 5)2 - y 2
f ) x 2 - 6x + 9 - 4y 2
NEL
a) Determine expressions for the possible
dimensions of these boxes.
b) Determine the dimensions and volume
of a box if x = 2 cm.
19. Determine the coordinates of the vertex
of each relation.
a) y = x 2 - 10x + 24
b) y = 2x 2 - 24x + 72
c) y = - 5x 2 + 500
d) y = 2x 2 - 7x - 4
e) y = 4x 2 + 16x
f ) y = x 2 + 10x + 25
Chapter 4
241
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4
Checklist
Process
✔ Question 4: Did you reflect
on your thinking to decide
which strategy you prefer?
✔ Questions 5 and 6: Did you
select strategies that are
appropriate for the
expressions?
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Chapter Self-Test
1. Determine the value of each symbol.
a) x 2 - 䉬x - 56 = (x + 7)(x - .)
b) 16x 2 - 䉬 = (. x - 3)( . x + 䊉)
c) 12x 2 + 䊉 x + 5 = (4x + .)(䉬x + 5 )
d) 25x 2 + 䊉 x + 49 = (. x + 䉬)2
2. Identify each trinomial that is modelled below, and state its factors.
a)
⫺x
⫺x
⫺x
⫺x
1 1 1
1 1 1
x2
x2
⫺x ⫺x ⫺x
x2
x2
⫺x ⫺x ⫺x
b)
✔ Question 9: Did you
connect factoring with
the factored form of
a quadratic relation from
Chapter 3?
2x2
⫺x
8x
⫺4
3. Factor each expression.
a) 20x 5 - 30x 3
b) –8yc 3 + 4y 2c - 6yc
c) 2a(3b + 5) + 7(3b + 5)
d) 2st + 6s + 5t + 15
4. a) Factor 25x 2 - 30x + 9 using two different strategies.
b) Which strategy do you prefer? Explain why.
5. Factor each expression.
a) x 2 + 4x - 77
b) a 2 - 3a - 10
c) 3x 2 - 12x + 12
d) m 3 + 3m 2 - 4m
6. Factor.
a) 6x 2 - x - 2
b) 8n 2 + 8n - 6
c) 9x 2 + 12x + 4
d) 6ax 2 + 5ax - 4a
7. A graphic arts company creates posters with areas that are given
RIEL H
HIG
TALE
8. Factor each expression.
FRIDAY
MAY 14
9. A parabola is defined by the equation y = 2x 2 - 11x + 5. Explain
NT
YM
CENTRAL G
NOON
by the equation A = 2x 2 + 11x + 12.
a) Write expressions for possible dimensions of the posters.
b) Write expressions for the dimensions of a poster whose width
is doubled and whose length is increased by 2. Write the new area
as a simplified polynomial.
c) Write expressions for possible dimensions of a poster whose area
is given by the expression 18x 2 + 99x + 108.
SHOW
a) 225x 2 - 4
b) 9a 2 - 48a + 64
c) x 6 - 4y 2
d) (3 + n)2 - 10(3 + n) + 25
how you can determine the vertex of the parabola without using
graphing technology.
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Chapter Self-Test
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Chapter Task
The Factoring Challenge
?
How well can you play this game to show what you know
about factoring polynomials?
Number of players: 3 per group
Materials needed for each player: 10 blank cards, a recording sheet,
a calculator (optional)
Rules
1. On your own, create 10 polynomials that satisfy the following conditions:
• Two must be quadratic binomials, but only one is factorable.
• Six must be quadratic trinomials, but only four are factorable.
• Two must have at least four terms, but only one is factorable.
Write each of your 10 polynomials on a separate card.
2. Form a group with two other students, and combine all your cards.
Place the cards face down so that the polynomials are not visible.
Decide on the order in which you will play. The goal of the game
is to accumulate the most points.
3. On your turn, choose a card and factor the polynomial if possible. If you
are correct, you get 5 points; if you are incorrect, you get 0 points. If you
say that the polynomial cannot be factored but it can, you get 0 points.
If the polynomial is not factorable, you must try to change one or two
coefficients or the constant to make it factorable. Your points are
determined by the number of changes you make. For example, suppose
that you turn over 2x 2 - 5x + 8. If you change it to 2x 2 - 10x + 8,
you get 1 point; if you change it to 2x 2 - 6x + 4, you get 2 points.
If you create another unfactorable polynomial, you get 0 points.
4. Take turns playing the game. You must record each polynomial you
turn over, its factors (if possible), and the points you get. If you turn
over a polynomial that cannot be factored, record the new polynomial
you create, its factors, and the points you get.
Recording Sheet for Shirley
Polynomial
Factors
169 ⫺ 4x2
(13 ⫹ 2x)(13 ⫺ 2x)
Task
Checklist
✔ Did you create a variety
of polynomials that met
the given conditions?
✔ Did you verify that the
Points Earned
5
169 ⫺ 4x2
correct number of
polynomials were factorable?
✔ Did you submit your
5. Each player gets 10 turns. The player with the most points wins.
recording sheet?
6. As a class, discuss any polynomials that were difficult to factor.
NEL
Chapter 4
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Chapter
5
Applying
Quadratic
Models
GOALS
You will be able to
• Investigate the y ⴝ a(x ⴚ h)2 ⴙ k form of
a quadratic relation
• Apply transformations to sketch graphs
of quadratic relations
• Apply quadratic models to solve
problems
• Investigate connections among the
different forms of a quadratic relation
An arch is a structure that spans
a distance and supports weight.
The ancient Romans were the first
people to use semicircular arches
in a wide range of structures.
The arches in this bridge are the
strongest type.
? What characteristics of these arches
suggest that they are parabolas?
NEL
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Getting Started
WORDS YOU NEED to Know
1. Match each term at the left with the correct description or example.
a) transformation
b) translation
c) reflection
d) parabola
e) vertex
f ) factored form of a
quadratic relation
i) a point that relates to the maximum or minimum value
of a quadratic relation
ii) the result of moving or changing the size of a shape according to a rule
iii) the result of sliding each point on a shape the same distance
in the same direction
iv) y = a(x - r)(x - s)
v) the result of flipping a shape to produce a mirror image of the shape
vi) the graph of a quadratic relation
SKILLS AND CONCEPTS You Need
Working with Transformations
Study
Aid
• For more help and practice,
see Appendix A-13.
4
-2
Apply the following transformations to ^ ABC shown at the left.
a) Translate ^ ABC 3 units right and 2 units up.
b) Reflect ^ ABC in the x-axis.
C
2
A
-4
EXAMPLE
y
6
B
0
2
-2
x
4
a)
y
6
C′
C
4
A′
B
B′
-2 A″
-2
B″ 4
6
-4
C″
2
A
246
Solution
Apply the same translation and the same reflection to points A, B, and C.
Plot the image points, and draw each image triangle.
6
-4
-4
Translations, reflections, rotations, and dilatations are types of
transformations. They can be applied to a point, a line, or a figure.
0
Getting Started
Original Point
Image Point
A(0, 1)
b)
Original Point
Image Point
A¿ (3, 3)
A(0, 1)
A– (0, 1)
B(2, 1)
B¿ (5, 3)
B(2, 1)
B– (2, 1)
C(2, 4)
C¿ (5, 6)
C(2, 4)
C– (2, 4)
x
2. a) In the diagram at the right, is figure B,
figure C, or figure D the result of a
translation of figure A? Explain.
b) Which figure is the result of a
reflection of figure A in the x-axis?
Explain.
c) Which figure is the result of a
reflection of figure A in the y-axis?
Explain.
y
B
A
2
x
-4
-2
0
-4
C
4
-2
D
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Getting Started
Understanding Quadratic Relations
A quadratic relation can be expressed algebraically as an equation in standard
form or factored form. You can determine information about its parabola from
the equation of the relation.
Study
Aid
• For more help and
practice, see Lesson 3.3,
Examples 2 to 4.
EXAMPLE
Determine the properties of the relation defined by the equation
y = (x - 2)(x - 4). Then sketch the graph of the relation.
Solution
The equation y = (x - 2)(x - 4) is the equation of a quadratic relation
in factored form.
The values x = 2 and x = 4 make the factors (x - 2) and (x - 4) equal
to zero. These are called the zeros of the relation and are the x-intercepts
of the graph.
The axis of symmetry is the perpendicular bisector of the line segment
2 + 4
or x = 3.
that joins the zeros. Its equation is x =
2
The vertex of the parabola lies on the axis of symmetry. To determine the
y-coordinate of the vertex, substitute x = 3 into the equation and evaluate.
y = (3 - 2)(3 - 4)
y = (1)( - 1)
y = -1
The vertex is (3, - 1).
y
6
y (x 2)(x 4)
x3
4
x-intercepts
2
x
0
-2
2
4 6
vertex
8
3. Determine the value of y in each quadratic relation for each value of x.
a) y = x 2 + 2x + 5, when x = - 4
b) y = x 2 - 3x - 28, when x = 7
4. Determine the zeros, the equation of the axis of symmetry, and
the vertex of each quadratic relation.
a) y = (x + 5)(x - 3)
b) y = 2(x - 4)(x + 1)
NEL
c) y = - 4x(x + 3)
Chapter 5
247
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• For help, see the Review of
Essential Skills and
Knowledge Appendix.
Question
Appendix
5, 6
A-13
y
8
-2
R Q′
2
0
R′
x
P′
Q″
-2
-4
Page 248
5. State the coordinates of the image point after applying the indicated
transformation(s).
a) A(3, 4) is translated 2 units left and 5 units up.
b) B(1, 5) is translated 4 units right and 3 units down.
c) C(2, 7) is translated 2 units left and 7 units up.
d) D(3, 5) is reflected in the x-axis.
6. Describe the transformation that was used to translate each triangle
6 Q
P
9:38 AM
PRACTICE
Aid
Study
5/8/09
6 8
R″
P″
onto the image in the diagram at the left.
a) ^PQR to ^P¿Q¿R¿
b) ^PQR to ^P–Q–R–
c) ^P¿Q¿R¿ to ^P–Q–R–
7. State the zeros, equation of the axis of symmetry, and the vertex
of each quadratic relation.
a)
b)
y
4
y
4
2
2
x
-4
-2
0
-2
2
4
x
-6 -4
-2
0
-2
-4
-4
-6
-6
-8
-8
-10
-10
2
8. Express each quadratic relation in standard form.
a) y = (x + 4)(x + 5)
b) y = (2x - 3)(x + 2)
c) y = - 3(x - 4)(x + 7)
d) y = (x + 5)2
9. Sketch the graph of each quadratic relation in question 8.
10. Express y = 2x 2 - 4x - 48 in factored form. Then determine
its minimum value.
11. Copy and complete the chart to show what you know about quadratic
relations. Share your chart with a classmate.
Definition:
Examples:
248
Getting Started
Special Properties:
Quadratic
Relation
Non-examples:
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Getting Started
APPLYING What You Know
YOU WILL NEED
Tiling Transformers
Jesse and Tyler decided to have a competition to see who could transform the
figures from the Start grid to the End grid in the fewest number of moves.
End
y
Start
y
8
6
• grid paper
• ruler
• coloured pencils or markers
• scissors
4
4
x
0
-8 -6 -4
4
6
8
x
-8 -6
-4
-2
0
-2
-4
2
?
What is the fewest number of moves needed to transform
these figures?
A.
On a piece of grid paper, draw the five figures on the Start grid. Cut out
each figure.
B.
Draw x- and y-axes on another piece of grid paper. Label each axis
from 10 to 10.
C.
Put the yellow figure on the grid paper in the position shown
on the Start grid.
D.
Apply one or more transformations to move the yellow figure
to the position shown on the End grid.
Make a table like the one below. Record the transformation(s) that
you used to move the yellow figure for part D.
E.
B(–3, 7)
y
8
6
4
B′(2, 1)
2
A(–6, 4)
x
-8 -6 -4
0
-2
2
4
Player 1
Move
1
Figure
yellow
Original
Coordinates
Transformation
New
Coordinates
A(–6, 4), B(–3, 7)
translation
5 units right,
6 units down
A(–1, –2), B(2, 1)
A′(–1, –2)
-4
2
F.
Repeat parts C to E for another figure. (The second figure can move
across the first figure if necessary.)
G.
Continue transforming figures and recording results until you have
created the design shown on the End grid.
H.
How many moves did you need? Compare your results with those
of other classmates. What is the fewest number of moves needed?
NEL
Chapter 5
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5.1
YOU WILL NEED
• graphing calculator
• dynamic geometry software,
or grid paper and ruler
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Stretching/Reflecting
Quadratic Relations
GOAL
Examine the effect of the parameter a in the equation
y ax 2 on the graph of the equation.
INVESTIGATE the Math
Suzanne’s mother checks the family’s investments regularly. When Suzanne
saw the stock chart that her mother was checking, she noticed trends in
sections of the graph. These trends looked like the shapes of the parabolas
she had been studying. Each “parabola” was a different shape.
As of 10-Jun-09
Canadian Dollars
80.00
75.00
70.00
65.00
60.00
Tech
Support
For help graphing relations and
adjusting the window settings
using a TI-83/84 graphing
calculator, see Appendix B-2
and B-4. If you are using a
TI-nspire, see Appendix B-38
and B-40.
250
?
What is the relationship between the value of a in the equation
y ax 2 and the shape of the graph of the relation?
A.
Enter y = x 2 as Y1 in the equation editor of a graphing calculator.
B.
The window settings shown are “friendly”
because they allow you to trace using
intervals of 0.1. Graph the parabola using
these settings.
C.
Enter y = 2x 2 in Y2 and y = 5x 2 in Y3,
and graph these quadratic relations. What
appears to be happening to the shape of the
graph as the value of a increases?
5.1 Stretching/Reflecting Quadratic Relations
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5.1
D.
E.
F.
Where would you expect the graph of y = 3x 2 to appear, relative to
the other three graphs? Check by entering y = 3x 2 into Y4 and graph
with a thick line. Was your conjecture correct?
1
1
Where would you expect the graphs of y = x 2 and y = x 2 to
2
4
appear, relative to the graph of y = x 2? Clear the equations from Y2,
1
1
Y3, and Y4. Enter y = x 2 into Y2 and y = x 2 into Y3, and graph
2
4
these quadratic relations. Describe the effect of the parameter a on the
parabola when 0 a 1.
3
Where you would expect the graph of y = x 2 to appear, relative to
4
3
the other three graphs? Check by entering y = x 2 into Y4 and graph
4
with a thick line.
G.
Clear the equations from Y2, Y3, and Y4. Enter y = - 4x 2 into Y2
1
and y = - x 2 into Y3, and graph these quadratic relations. Describe
4
the effect of a on the parabola when a 0.
H.
Ask a classmate to give you an equation in the form y = ax 2, where
a 0. Describe to your classmate what its graph would look like
relative to the other three graphs. Verify your description by graphing
the equation in Y4.
I.
How does changing the value of a in the equation y = ax 2 affect
the shape of the graph?
Tech
Support
Move the cursor to the left of
Y4. Press ENTER
to
change the line style to make
the line thick.
parameter
a coefficient that can be
changed in a relation; for
example, a, b, and c are
parameters in y ax2 bx c
Reflecting
J.
Which parabola in the stock chart has the greatest value of a? Which
has the least value of a? Which parabolas have negative values of a?
Explain how you know.
K.
What happens to the x-coordinates of all the points on the graph
of y = x 2 when the parameter a is changed in y = ax 2? What
happens to the y-coordinates? What happens to the shape of the
parabola near its vertex?
L.
NEL
State the ranges of values of a that will cause the graph of y = x 2 to be
i) vertically stretched
ii) vertically compressed
iii) reflected across the x-axis
vertical stretch
a transformation that increases
all the y-coordinates of a
relation by the same factor
vertical compression
a transformation that decreases
all the y-coordinates of a
relation by the same factor
Chapter 5
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APPLY the Math
EXAMPLE 1
Selecting a transformation strategy
to graph a parabola
a) Sketch the graph of the equation y = 3x 2 by transforming the graph
of y = x 2.
b) Describe how the graphs of y = 3x 2 and y = - 3x 2 are related.
Zack’s Solution
a)
2 1
x
y
4
1
0
1
2
0
1
4
y
12
I created a table of values to
determine five points on the
graph of y x2.
y x2
I plotted the points on a grid and
joined them with a smooth curve.
10
8
6
4
2
x
-8 -6 -4
-2
0
2
-2
4
8
6
I decided to call this my five-point
sketch.
-4
x
2
1
0
1
2
y
12
3
0
3
12
(2, 4)
8
6
(2, 4)
4
2
x
-8 -6 -4
-2
0
-2
-4
252
5.1 Stretching/Reflecting Quadratic Relations
2
4
6
(2, 12)
43
(2, 12)
10
y 3x2
To transform my graph into
a graph of y 3x2, I multiplied
the y-coordinates of each point on
y x2 by 3. For example,
y
12
y x2
I can use these five points any
time I want to sketch the graph
of y x2 because they include
the vertex and two points on
each side of the parabola.
8
I plotted and joined my new
points to get the graph of
y 3x2. a 3 represents a
vertical stretch by a factor of 3.
This means that the y-coordinates
of the points on the graph of
y 3x2 will become greater
faster, so the parabola will be
narrower near its vertex compared
to the graph of y x2.
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5.1
b)
2 1
0
y 12 3
0
x
1
3 12
y
12
10
8
6
y 3x2
4
2
x
-4
-2
0
-2
-4
2
2
4
y 3x2
To get the graph of y 3x2,
I multiplied the y-coordinates of
all the points on the graph of
y 3x2 by 1. For example,
(2,12)
(2,12)
12 (1)
a 3 represents a vertical
stretch by a factor of 3 and a
reflection in the x-axis. This means
that all the points on the graph of
y 3x2 are reflected in the x-axis.
-6
-8
-10
-12
(2, -12)
The graph of y = - 3x 2 is
the reflection of the graph of
y = 3x 2 in the x-axis.
EXAMPLE 2
Connecting the value of a to a graph
Determine an equation of a quadratic relation that models the arch
of San Francisco’s Bay Bridge in the photograph below.
NEL
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Mary’s Solution: Representing the picture on a hand-drawn grid
y
2
x
-8
-6
-4
-2
2
4
6
8
-2
I used a photocopy of the photograph. I laid
a transparent grid with axes on top of the
photocopy.
I located a point on the graph and estimated
the coordinates of the point to be (5, 1).
I placed the origin at the vertex of the arch. I did
this since all parabolas defined by y ax2 have
their vertex at (0, 0).
y = ax 2
1 = a(5)2
1 = 25a
25a
1
=
25
25
1
= a
25
The equation of the graph is in the form y ax2.
To determine the value of a, I had to determine the
coordinates of a point on the parabola. I chose the
point (5, 1). I substituted x 5 and y 1 into the
equation and solved for a.
An equation that models the arch of the bridge
1 2
is y =
x .
25
The graph that models the arch is a vertical
compression of the graph of y x2 by a factor
1
of
.
25
Sandeep’s Solution: Selecting dynamic geometry software
4
2
-8
-6
-4
-2
2
4
6
8
-2
I imported the photograph into dynamic geometry
software. I superimposed a grid over the
photograph. Then I adjusted the grid so that the
origin was at the vertex of the bridge’s parabolic
arch. I need to create a graph using the relation
y = ax2 by choosing a value for the parameter a.
-4
Tech
Support
For help creating and graphing
relations using parameters in
dynamic geometry software,
as well as animating the
parameter, see Appendix B-17.
254
5.1 Stretching/Reflecting Quadratic Relations
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5.1
f (x) = a •x2
a = 1.00
4
2
-8
-6
-4
-2
2
4
6
8
-2
When I used a 1, the graph of y = x 2 appeared.
The parabola was too narrow. It had to be vertically
compressed to fit the arch. To do this, I needed
a lower value of a, between 0 and 1. I needed
a positive value because the arch opens upward.
-4
f (x) = a •x2
a = 0.04
I tried a 0.5, but the parabola was not wide
enough.
4
2
-8
-6
-4
-2
2
4
6
8
-2
-4
An equation that models the bridge is y = 0.04x 2.
I tried a 0.1. This value gave me a better fit.
I still wasn’t satisfied, so I tried different values of a
between 0 and 0.1. I found that a 0.04 gave me
a good fit.
Vertically compressing the graph of y = x 2 by a factor
of 0.04 creates a graph that fits the photograph.
In Summary
Key Idea
• When compared with the graph of y x2, the graph of y ax2 is
a parabola that has been stretched or compressed vertically by a factor of a.
Need to Know
• Vertical stretches are determined by the value of a. When a 1, the
graph is stretched vertically. When a 1, the graph is stretched
vertically and reflected across the x-axis.
• Vertical compressions are also determined by the value of a. When
0 a 1, the graph is compressed vertically. When 1 a 0,
the graph is compressed vertically and reflected across the x-axis.
• If a 0, the parabola opens upward.
• If a 0, the parabola opens downward.
y ax2,
a1
y x2
x
y ax2,
1 a 0
NEL
y
y ax2,
0a1
y ax2,
a 1
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CHECK Your Understanding
1. Match each graph with the correct equation. The graph of y = x2
is in green in each diagram.
2 2
x
3
d) y = - 0.4x 2
a) y = 4x 2
c)
b) y = - 3x 2
i)
ii)
y
10
4
8
y x2
6
-4
x
0
-2
-2
2
4
0
-2
2
4
8
6
2
x
-4
-2
0
-2
-4
-4
-6
-6
y
10
4
x
y x2 2
-2
iv)
y
6
y x2
2
4
-4
iii)
y
6
y =
2
4
y x2
4
2
x
-4
-2
0
-2
2
4
2. The graph of y = x 2 is transformed to y = ax 2 (a Z 1). For each
point on y = x 2, determine the coordinates of the transformed point
for the indicated value of a.
a) (1, 1), when a = 5
c) (5, 25), when a = - 0.6
1
b) (2, 4), when a = - 3
d) (4, 16), when a =
2
3. Write the equations of two different quadratic relations that match
each description.
a) The graph is narrower than the graph of y = x 2 near its vertex.
b) The graph is wider than the graph of y = - x 2 near its vertex.
c) The graph opens downward and is narrower than the graph
of y = 3x 2 near its vertex.
PRACTISING
4. Sketch the graph of each equation by applying a transformation
K
256
to the graph of y = x 2. Use a separate grid for each equation, and
start by sketching the graph of y = x 2.
1
a) y = 3x 2
d) y = x 2
4
3
b) y = - 0.5x 2
e) y = - x 2
2
2
c) y = - 2x
f ) y = 5x 2
5.1 Stretching/Reflecting Quadratic Relations
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5.1
5. Describe the transformation(s) that were applied to the graph of y = x 2
to obtain each black graph. Write the equation of the black graph.
a)
c)
y
14
y x2
y
y x2
4
12
2
10
x
8
-8 -6 -4
-2
6
0
2
4
6
d)
6
8
4
6
4
y x2
x
-2
y
10
2
-8 -6 -4
8
-10
8
y
8
y x2
6
-8
x
b)
4
-6
2
-2
2
-2
-4
4
-8 -6 -4
0
0
2
-2
4
6
2
x
8
-8 -6 -4
-2
0
-4
-2
-6
-4
2
4
6
8
6. Andy modelled the arch of the bridge in the photograph at the right
y
by tracing a parabola onto a grid. Now he wants to determine an
equation of the parabola. Explain the steps he should use to do this,
and state the equation.
2
7. Determine an equation of a quadratic model for each natural arch.
0
C
a) Isle of Capri in Italy
b) Corona Arch in Utah
x
-4
NEL
-2
0
2
4
6
x
-6
-4
-2
0
-2
-2
-4
-4
-6
-6
2
x
-4
-2
2
4
-2
y
y
y x2
-4
4
Chapter 5
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8. Identify the transformation(s) that must be applied to the graph
of y = x 2 to create a graph of each equation. Then state the
coordinates of the image of the point (2, 4).
a) y = 4x 2
c) y = 0.25x 2
e) y = - x 2
2
1
b) y = - x 2
d) y = - 5x 2
f) y = x 2
3
5
9. By tracing the bridge at the left onto a grid, determine an equation that
A
models the lower outline of the Sydney Harbour Bridge in Australia.
10. Seth claims that changing the value of a in quadratic relations of the
T
form y = ax 2 will never result in a parabola that is congruent to the
parabola y = x 2. Do you agree or disagree? Justify your decision.
11. Copy and complete the following table.
Equation
Direction
of Opening
(upward/
downward)
Description of
Transformation
(stretch/
compress)
Shape of Graph
Compared with
Graph of y x2
(wider/narrower)
y = 5x 2
y = 0.25x 2
y = -
1 2
x
3
y = - 8x 2
12. Explain why it makes sense that each statement about the graph
of y = ax 2 is true.
a) If a 0, then the parabola opens downward.
b) If a is a rational number between 1 and 1, then the parabola
is wider than the graph of y = x 2.
c) The vertex is always (0, 0).
Extending
13. The graph of y = ax 2 (a Z 1, a 0) is either a vertical stretch or
a vertical compression of the graph of y = x 2. Use graphing technology
to determine whether changing the value of a has a similar effect on the
1
graphs of equations such as y = ax, y = ax 3, y = ax 4, and y = ax 2.
14. The equation of a circle with radius r and centre (0, 0) is
x 2 + y 2 = r 2.
a) Explore the effect of changing positive values of a when graphing
ax 2 + ay 2 = r 2.
b) Explore the effects of changing positive values of a and b
when graphing ax 2 + by 2 = r 2.
258
5.1 Stretching/Reflecting Quadratic Relations
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Exploring Translations
of Quadratic Relations
GOAL
YOU WILL NEED
Investigate the roles of h and k in the graphs of y x 2 k,
y (x h) 2, and y (x h) 2 k.
• grid paper
• ruler
• graphing calculator
EXPLORE the Math
Hammad has been asked to paint
a mural of overlapping parabolas on
a wall in his school. A sketch of his
final design is shown at the right.
He is using his graphing calculator
to try to duplicate his design. His
design uses parabolas that have the
same shape as y = x 2, but he
doesn’t know what equations he
should enter into his graphing
calculator to place the parabolas in
different locations on the screen.
y
8
6
4
2
x
-8
-6 -4
-2
0
-2
2
4
6
-4
-6
-8
?
What is the connection between the location of the vertex
of a parabola and the equation of its quadratic relation?
A.
Enter the equation y = x 2 as Y1 in the
equation editor of a graphing calculator.
Graph the equation using the window
settings shown.
B.
C.
Tech
Support
For help graphing relations,
changing window settings,
and tracing along a graph
using a TI-83/84 graphing
calculator, see Appendix B-2
and B-4. If you are using a
TI-nspire, see Appendix B-38
and B-40.
Enter an equation of the form y = x 2 + k
in Y2 by adding or subtracting a number after the x 2 term. For example,
y = x 2 + 1 or y = x 2 - 3. Graph your equation, and compare
the graph with the graph of y = x 2. Try several other equations,
replacing the one you have in Y2 each time. Be sure to change
the number you add or subtract after the x 2 term.
Tech
Support
Copy this table. Use the table to record your findings for part B.
Use the
TRACE
Value of k
0
NEL
8
Equation
Distance and Direction from y x2
Vertex
y x2
not applicable
(0, 0)
key and
the up arrow
to help
you distinguish one graph
from another.
Chapter 5
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Investigate what happens to the graph of y = x 2 when a number is
added to or subtracted from the value of x before it is squared, creating
an equation of the form y = (x - h)2. For example, y = (x + 1)2 or
y = (x - 2)2. Graph your new equations in Y2 each time using
a graphing calculator. Then copy this table and record your findings.
D.
Value of h
0
Equation
Distance and Direction from y x2
Vertex
y x2
not applicable
(0, 0)
E.
Identify the type of transformations that have been applied to the
graph of y = x 2 to obtain the graphs in your table for part C and
your table for part D.
F.
Make a conjecture about how you could predict the equation of a parabola
if you knew the translations that were applied to the graph of y = x 2.
G.
Copy and complete this table to investigate and test your conjecture
for part F.
Value Value
of h of k
0
0
Relationship to y ⴝ x 2
Equation
Left/Right
y = x2
not applicable not applicable
left 3
4
1
Up/Down
Vertex
(0, 0)
down 5
y = (x - 4)2 + 1
(2, 6)
y = (x + 5)2 - 3
H.
Use what you have discovered to identify the equations that Hammad
should type into his calculator to graph the parabolas in the mural design.
I.
If the equation of a quadratic relation is given in the form
y = (x - h)2 + k, what can you conclude about its vertex?
Reflecting
260
J.
Describe how changing the value of k in y = x 2 + k affects
i) the graph of y = x 2
ii) the coordinates of each point on the parabola y = x 2
iii) the parabola’s vertex and axis of symmetry
K.
Describe how changing the value of h in y = (x - h)2 affects
i) the graph of y = x 2
ii) the coordinates of each point on the parabola y = x 2
iii) the parabola’s vertex and axis of symmetry
5.2 Exploring Translations of Quadratic Relations
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5.2
L.
For parabolas defined by y = (x - h)2 + k,
i) how do their shapes compare to the parabola defined by y = x 2?
ii) what is the equation of the axis of symmetry?
iii) what are the coordinates of the vertex?
In Summary
Key Ideas
• The graph of y = (x - h)2 + k is congruent to the graph of y = x 2, but
translated horizontally and vertically.
• Translations can also be described as shifts. Vertical shifts are up or
down, and horizontal shifts are left or right.
Need to Know
• The value of h tells how far and in what
direction the parabola is translated
horizontally. If h 6 0, the parabola is
translated h units left. If h 7 0, the
parabola is translated h units right.
• The vertex of y = (x - h)2 is the point (h, 0).
• The equation of the axis of symmetry of
y = (x - h)2 is x = h.
y (x h)2
x
h0
• The value of k tells how far and in what
direction the parabola is translated vertically.
If k 6 0, the parabola is translated k units
down. If k 7 0, the parabola is translated
k units up.
• The vertex of y = x 2 + k is the point (0, k).
• The equation of the axis of symmetry of
y = x 2 + k is x = 0.
• The vertex of y = (x - h)2 + k is the
point (h, k).
• The equation of the axis of symmetry
of y = (x - h)2 + k is x = h.
h0
y
h0
y x2 k
y
k0
k0
x
k0
y x2
y (x h)2 k
y
(h, k)
x
xh
NEL
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FURTHER Your Understanding
1. The following transformations are applied to a parabola with
the equation y = x 2. Determine the values of h and k, and write
the equation in the form y = (x - h)2 + k.
a) The parabola moves 3 units right.
b) The parabola moves 4 units down.
c) The parabola moves 2 units left.
d) The parabola moves 5 units up.
e) The parabola moves 7 units down and 6 units left.
f ) The parabola moves 2 units right and 5 units up.
2. Match each equation with the correct graph.
a) y = (x - 2)2 + 3
b) y = (x + 2)2 - 3
i)
y
8
iii)
c) y = (x + 3)2 - 2
d) y = (x - 3)2 + 2
y
8
v)
6
6
4
4
4
2
2
2
x
-8 -6 -4 -2
ii)
-4
2
-2
y
6
iv)
2
x
-8 -6 -4 -2
-2
-4
-2
0
2
-2
4
6
vi)
6
4
4
2
2
x
2
x
0
-2
-2
y
6
2
-2
0
-4
y
8
4
0
-8 -6 -4 -2
x
x
0
y
6
2
4
6
8
-4
-2
0
-2
2
4
6
-4
3. Sketch the graph of each relation by hand. Start with the graph
of y = x 2, and apply the appropriate transformations.
a) y = x 2 - 4
b) y = (x - 3)2
c) y = x 2 + 2
d) y = (x + 5)2
e) y = (x + 1)2 - 2
f ) y = (x - 5)2 + 3
4. Describe the transformations that are applied to the graph of y = x 2
to obtain the graph of each quadratic relation.
a) y = x 2 + 5
c) y = - 3x 2
b) y = (x - 3)2
d) y = (x + 7)2
1 2
x
2
f ) y = (x + 6)2 + 12
e) y =
5. State the vertex and the axis of symmetry of each parabola in question 4.
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5.2 Exploring Translations of Quadratic Relations
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Graphing Quadratics
in Vertex Form
YOU WILL NEED
GOAL
Graph a quadratic relation in the form y = a(x - h)2 + k by using
transformations.
• grid paper
• ruler
LEARN ABOUT the Math
Srinithi and Kevin are trying to sketch the graph of the quadratic relation
y = 2(x - 3)2 - 8 by hand. They know that they need to apply a series
of transformations to the graph of y = x 2.
?
How do you apply transformations to the quadratic relation
y x2 to sketch the graph of y = 2(x - 3)2 - 8?
EXAMPLE 1
Selecting a transformation strategy
to graph a quadratic relation
Use transformations to sketch the graph of y = 2(x - 3)2 - 8.
Srinithi’s Solution: Applying a horizontal translation first
y = x2
x
2
1
0
1
2
y
4
1
0
1
4
y = (x - 3)2
x
1
2
3
4
5
y
4
1
0
1
4
y x2
Since h 3, I added 3 to the
x-coordinate of each point on
y = x 2. This means that the vertex
is (3, 0).
vertex form
a quadratic relation of the form
y = a(x - h)2 + k, where the
vertex is (h, k)
y (x 3)2
y
8
I began by graphing y = x 2 using
five key points. The quadratic
relation y = 2(x - 3)2 - 8 is
expressed in vertex form.
The equation of the new red
graph is y = (x - 3)2. To draw it,
I translated the green parabola
3 units right.
6
4
2
x
-4
-2
0
-2
2
4
6
-4
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y = 2(x - 3)2
x
1
2
3
4
5
y
8
2
0
2
8
y 2(x 3)2
y
8
y (x 3)2
6
4
2
x
-4
-2
0
2
-2
4
Since a = 2, I multiplied all the
y-coordinates of the points on the
red graph by 2. The vertex stays at
(3, 0). The equation of this graph
is y = 2(x - 3)2.
To draw this new blue graph,
I applied a vertical stretch by
a factor of 2 to the red graph.
The blue graph looks correct
because the graph with the
greater a value should be
narrower than the other graph.
6
-4
y = 2(x - 3)2 - 8
x
1
2
3
4
5
y
0
6 8
6
0
4
y 2(x 3)2
2
y 2(x 3)2 8
y
x
-4
-2
0
2
-2
4
6
I knew that k = - 8. I subtracted
8 from the y-coordinate of each
point on the blue graph. The
vertex is now (3, 8). The
equation of the graph is
y = 2(x - 3)2 - 8.
Since k 6 0, I knew that I had to
translate the blue graph 8 units
down to get the final black graph.
-4
-6
-8
Kevin’s Solution: Applying a vertical stretch first
y
8
6
y x2
y 2x2
4
2
x
-4
-2
0
-2
2
4
6
Since a 2, I decided to stretch
the graph of y = x 2 vertically by
a factor of 2. I multiplied the
y-coordinate of each point on
the graph of y = x 2 by 2.
The equation of the resulting red
graph is y = 2x 2.
-4
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5.3
y 2x2
I applied both translations in one
step. Adding 3 to the x-coordinate
and subtracting 8 from the
y-coordinate from each point on
the red graph causes the red
graph to move 3 units right
and 8 units down.
y
4
2
x
-4
-2
0
2
-2
4
-4
6
y 2(x 3)2 8
-6
-8
The equation of the resulting
black graph is y = 2(x - 3)2 - 8.
(3, –8)
Reflecting
A.
Why was it not necessary for Kevin to use two steps for the
translations? In other words, why did he not have to shift the graph
to the right in one step, and then down in another step?
B.
What are the advantages and disadvantages of each solution?
C.
How can thinking about the order of operations applied to the
coordinates of points on the graph of y = x 2 help you apply
transformations to draw a new graph?
APPLY the Math
EXAMPLE 2
Reasoning about sketching the graph of a quadratic relation
Sketch the graph of y = - 3(x + 5)2 + 1, and explain your reasoning.
Winnie’s Solution: Connecting a sequence of transformations to the equation
Applying a vertical stretch of factor 3 and a reflection
in the x-axis gives the graph of y = - 3x 2.
y
6
y 3x2
In the quadratic relation y = - 3(x + 5)2 + 1, the
value of a is 3. This represents a vertical stretch
by a factor of 3 and a reflection in the x-axis.
y x2
4
2
x
-6 -4
-2
0
-2
2
4
6
I noticed that I can combine the stretch and
reflection into a single step by multiplying each
y-coordinate of points on y = x2 by 3.
-4
-6
y 3x2
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In the equation, h = - 5 and k = 1. Therefore,
the vertex is at ( - 5, 1). I translated the blue
graph 5 units left and 1 unit up.
I determined that the vertex is (5, 1). Then I
shifted all the points on the graph of y = - 3x 2
so that they were 5 units left and 1 unit up.
y
y x2
6
4
I drew a smooth curve through the new points
to sketch the graph.
2
x
-6 -4
-2
0
2
-2
4
6
-4
-6
y 3x2
y 3(x 5)2 1
Beth’s Solution: Connecting the properties of a parabola to the equation
Based on the equation y = - 3(x + 5)2 + 1,
the parabola has these properties:
• Since a 6 0, the parabola opens downward.
• The vertex of the parabola is (5, 1).
• The equation of the axis of symmetry is x = - 5.
y = - 3(- 3 + 5)2 + 1
y = - 3(2)2 + 1
y = - 12 + 1
y = - 11
Since the equation was given in vertex form, I listed
the properties of the parabola that I could determine
from the equation.
To determine another point on the parabola, I let
x 3.
Therefore, (3, 11) is a point on the parabola.
y
2
x
-8 -6 -4
-2
0
-2
2
4
6
8
-4
-6
-8
I plotted the vertex and the point I had determined,
(3, 11). Then I drew the axis of symmetry. I
used symmetry to determine the point directly
across from (3, 11). This point is ( 7, 11).
I plotted the points and joined them with
a smooth curve.
-10
-12
x 5
-14
y 3(x 5)2 1
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5.3 Graphing Quadratics in Vertex Form
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5.3
EXAMPLE 3
Reasoning about the effects of transformations on a quadratic relation
For a high school charity event, the principal pays to drop a watermelon
from a height of 100 m. The height, h, in metres, of the watermelon after
t seconds is h = - 0.5gt 2 + k, where g is the acceleration due to gravity
and k is the height from which the watermelon is dropped.
On Earth, g = 9.8 m/s2.
a) The clock that times the fall of the watermelon runs for 3 s before the
principal releases the watermelon. How does this change the graph
shown? Determine the equation of the new relation.
b) On Mars, g = 3.7 m/s2. Suppose that an astronaut dropped
a watermelon from a height of 100 m on Mars. Determine the
equation for the height of the watermelon on Mars. How does the
graph for Mars compare with the graph for Earth for part a)?
h
100
80
60
40
20
t
-10
0
10
h 4.9t2 100, t 0
c) The principal drops another watermelon from a height of 50 m on Earth.
How does the graph for part a) change? How does the relation change?
d) Repeat part c) for an astronaut on Mars.
Nadia’s Solution
a) The equation of the original relation is
h = - 0.5(9.8)t 2 + 100
h = - 4.9t 2 + 100, where t Ú 0
The parabola is translated 3 units right.
The equation of the new relation is
h = - 4.9(t - 3)2 + 100, where t Ú 3.
h
The original graph is a parabola that opens
downward, with vertex (0, k) (0, 100). I wrote
and simplified the original relation. Only the right
branch of the parabola makes sense in this
situation since time can’t be negative.
I subtracted 3 from the t-coordinate to determine
the new relation. Since the watermelon is not
falling before 3 s, the relation only holds for t Ú 3.
If the clock runs for 3 s before the watermelon is
dropped, then the watermelon will be at its highest
point at 3 s. So, the vertex of the new parabola is
(3, 100), which is a shift of the original parabola
3 units right.
100
80
60
40
20
t
-10
0
10
h 4.9(t 3)2 100, t 3
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b) The equation of the relation on Mars is
h = - 0.5(3.7)t 2 + 100
h = - 1.85t 2 + 100, where t Ú 0
The graph for Mars is wider near the vertex.
I used the value of g on Mars, g = 3.7m/s2,
instead of g = 9.8 m/s2.
h
100
80
A lesser (negative) a-value means that
the parabola is wider.
60
40
20
t
0
-10
10
h 1.85t2 100, t 0
The t-intercept is farther from the origin, so the
watermelon would take longer to hit the ground
on Mars compared to Earth.
c) The equation of the new relation is
h = - 4.9t 2 + 50, where t Ú 0.
In the relation, k changes from 100 to 50.
The new graph has the same shape but is
translated 50 units down.
h
60
The new vertex is half the distance above the
origin, at (0, 50) instead of (0, 100). This is a shift
of 50 units down.
40
20
t
0
-10
10
h 4.9t2 50, t 0
d) The new graph for Mars is wider than the
original graph and is translated 50 units down.
h
60
40
20
t
-10
0
The new graph for Mars is wider than the original
graph, like the graph for part b). It is translated
down, like the graph for part c).
10
h 1.85t2 50, t 0
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5.3 Graphing Quadratics in Vertex Form
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5.3
In Summary
Key Idea
• Compared with the graph of y = x 2, the graph of y = a(x - h)2 + k is
a parabola that has been stretched or compressed vertically by a factor
of a, translated horizontally by h, and translated vertically by k. As well,
if a 6 0, the parabola is reflected in the x-axis.
y a(x h)2 k
y
Need to Know
• The vertex of y = a(x - h)2 + k has the coordinates (h, k). The equation
of the axis of symmetry of y = a(x - h)2 + k is x = h.
• When sketching the graph of y = a(x - h)2 + k as a transformation
of the graph of y = x 2, follow the order of operations for the
arithmetic operations to be performed on the coordinates of each point.
Apply vertical stretches/compressions and reflections, which involve
multiplication, before translations, which involve addition or subtraction.
x
y x2
y ax2
(h, k)
xh
CHECK Your Understanding
1. Describe the transformations you would apply to the graph of y = x 2,
in the order you would apply them, to obtain the graph of each
quadratic relation.
1
a) y = x 2 - 3
c) y = - x 2
2
2
b) y = (x + 5)
d) y = 4(x + 2)2 - 16
2. For each quadratic relation in question 1, identify
i) the direction in which the parabola opens
ii) the coordinates of the vertex
iii) the equation of the axis of symmetry
3. Sketch the graph of each quadratic relation. Start with a sketch of y = x 2,
and then apply the appropriate transformations in the correct order.
a) y = (x + 5)2 - 4
c) y = 2(x - 3)2
1
b) y = - 0.5x 2 + 8
d) y = (x - 4)2 - 2
2
PRACTISING
4. What transformations would you apply to the graph of y = x 2
to create the graph of each relation? List the transformations
in the order you would apply them.
a) y = - x 2 + 9
c) y = (x + 2)2 - 1
2
b) y = (x - 3)
d) y = - x 2 - 6
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1
e) y = - 2(x - 4)2 + 16
g) y = - (x + 4)2 - 7
2
1
2
f ) y = (x + 6) + 12
h) y = 5(x - 4)2 - 12
2
5. Sketch a graph of each quadratic relation in question 4 on a separate
grid. Use the properties of the parabola and additional points as
necessary.
6. Match each equation with the correct graph.
1
(x - 2)2 - 5
2
1
b) y = (x - 4)2 - 2
2
d) y = - 2(x - 2)2 - 5
a) y =
e) y = 4(x - 5)2 - 2
c) y = - 2(x + 2)2 + 5
i)
iii)
y
8
6
4
4
2
0
-4
2
-2
4
6
-4
ii)
y
6
4
2
-2
0
-2
-2
-6
-6
y
6
4
2
2
-2
4
6
vi)
-2
6
8
y
2
-4
0
-2
4
x
x
-6 -4
2
-2
-4
iv)
2
0
-4
4
0
x
4
2
-2
x
-4
y
6
x
x
-2
1 2
x - 2
3
v)
y
6
2
-4
f) y =
2
4
-2
0
-2
2
4
6
-4
-6
-4
-4
-8
-6
-6
-10
7. Sketch the graph of each quadratic relation by hand. Start with
a sketch of y = x 2, and then apply the appropriate transformations
in the correct order.
3
a) y = - (x - 2)2
d) y = x 2 - 5
4
1
1
b) y = (x + 2)2 - 8
e) y = (x - 2)2 - 5
2
2
c) y = - 3(x - 1)2 + 7 f ) y = - 1.5(x + 3)2 + 10
270
5.3 Graphing Quadratics in Vertex Form
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5.3
8. Copy and complete the following table.
Quadratic
Relation
Stretch/
Compression
Factor
3
Reflection
in the x-axis
no
Horizontal/
Vertical
Translation
Vertex
Axis of
Symmetry
right 2,
down 5
(2, 5)
x2
y = 4(x + 2)2 - 3
y = - (x - 1)2 + 4
y = 0.8(x - 6)2
y = 2x 2 - 5
9. Determine the equations of three different parabolas with a vertex
C
at (2, 3). Describe how the graphs of the parabolas are different
from each other. Then sketch the graphs of the three relations on
the same set of axes.
10. When an object with a parachute is released to fall freely, its height, h, in
metres, after t seconds is modelled by h = - 0.5(g - r)t 2 + k, where g
is the acceleration due to gravity, r is the resistance offered by the
parachute, and k is the height from which the object is dropped. On
Earth, g 9.8 m/s 2. The resistance offered by a single bed sheet is
0.6 m/s 2, by a car tarp is 2.1 m/s 2, and by a regular parachute is 8.9 m/s 2.
a) Describe how the graphs will differ for objects dropped from a height
of 100 m using each of the three types of parachutes.
b) Is it possible to drop an object attached to the bed sheet and a similar
object attached to a regular parachute and have them hit the ground
at the same time? Describe how it would be possible and what the
graphs of each might look like.
11. Write the equation of a parabola that matches each description.
a) The graph of y = x 2 is reflected about the x-axis and then
translated 5 units up.
b) The graph of y = x 2 is stretched vertically by a factor of 5 and
then translated 2 units left.
1
c) The graph of y = x 2 is compressed vertically by a factor of
5
and then translated 6 units down.
d) The graph of y = x 2 is reflected about the x-axis, stretched
vertically by a factor of 6, translated 3 units right, and translated
4 units up.
12. Sketch the graph of each parabola described in question 11 by applying
K
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the given sequence of transformations. Use a separate grid for each graph.
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6
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reasoning.
2
c) y = - (x - 3)2 + 5
3
2
b) y = - (x - 3)2 + 5 d) y = (x - 3)2 + 5
3
14. A sky diver jumped from an airplane. He used his watch to time the
length of his jump. His height above the ground can be modelled by
h = - 5(t - 4)2 + 2500, where h is his height above the ground in
metres and t is the time in seconds from the time he started the timer.
a) How long did the sky diver delay his jump?
b) From what height did he jump?
a) y = -
2
x
0
-2
9:35 AM
13. Which equation represents the graph shown at the left? Explain your
4
-2
5/8/09
2
4
6
8
2 2
x + 5
3
15. A video tracking device recorded the height, h, in metres, of a baseball
A
Safety Connection
A helmet and goggles are
important safety equipment
for skydivers.
after it was hit. The data collected can be modelled by the relation
h = - 5(t - 2)2 + 21, where t is the time in seconds after the ball
was hit.
a) Sketch a graph that represents the height of the baseball.
b) What was the maximum height reached by the baseball?
c) When did the baseball reach its maximum height?
d) At what time(s) was the baseball at a height of 10 m?
e) Approximately when did the baseball hit the ground?
16. When a graph of y = x 2 is transformed, the point (3, 9) moves to
T
(8, 17). Describe three sets of transformations that could make this
happen. For each set, give the equation of the new parabola.
17. Express the quadratic relation y = 2(x - 4)(x + 10) in both standard
form and vertex form.
18. Copy and complete the chart to show what you know about the
quadratic relation y = - 2(x + 3)2 + 4.
Translation:
Reflection:
Stretch/
Compression:
y 2(x 3)2 4
Vertex:
Extending
19. Determine one of the zeros of the quadratic relation
y = ax 272
5.3 Graphing Quadratics in Vertex Form
k 2
(k - 2)2
.
b 2
4
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
A:
How do you know whether the graph of y ax2 will have
a wider or narrower shape near its vertex, compared
with the graph of y x 2?
The shape depends on the value of a in the equation. Each y-value
is multiplied by a factor of a. When a 1, the y-values increase.
The parabola appears to be vertically stretched and becomes narrower
near its vertex. When 0 a 1, the y-values decrease. The parabola
appears to be vertically compressed and becomes wider near its vertex.
Q:
Why is the vertex form, y a(x h)2 k, useful
for graphing quadratic relations?
A1:
You can use the constants a, h, and k to determine how the graph
of y = x 2 has been transformed.
• When a 1, the parabola is vertically stretched and
when 0 a 1, the parabola is vertically compressed.
• When a 0, the parabola is reflected in the x-axis.
• The parabola is translated to the right when h 0 and to the left
when h 0. The parabola is translated up when k 0 and down
when k 0.
• The coordinates of the vertex are (h, k).
A2:
Study
Aid
• See Lesson 5.1,
Examples 1 and 2.
• Try Mid-Chapter Review
Questions 1 and 2.
Study
Aid
• See Lesson 5.3,
Examples 1 to 3.
• Try Mid-Chapter Review
Questions 3, 4, and 6 to 8.
You can use the constants a, h, and k to determine key features
of the parabola.
• When a 0, the parabola opens upward. When a 0, the
parabola opens downward.
• The coordinates of the vertex are (h, k).
• The equation of the axis of symmetry is x = h.
You can use these properties, as well as the coordinates of a few other
points, to draw an accurate sketch of any parabola.
Q:
When you use transformations to sketch a graph, why is the
order in which you apply the transformations important?
A:
When a graph is transformed, operations are performed on the
coordinates of each point. Apply transformations in the same order you
would apply calculations. Apply vertical stretches/compressions and
reflections (multiplication) before translations (addition or subtraction).
Stretch
NEL
Reflect
Study
Aid
• See Lesson 5.3, Examples
1 to 3.
• Try Mid-Chapter Review
Question 5.
Translate
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PRACTICE Questions
Lesson 5.1
4. These parabolas were
1. Sketch the graph of each equation by correctly
applying the required transformation(s) to
points on the graph of y = x 2. Use a separate
grid for each graph.
a) y = 2x 2
c) y = - 3x 2
2
b) y = - 0.25x 2
d) y = x 2
3
2. Describe the transformation(s) that were applied
to the graph of y = x 2 to obtain each black
graph. Write the equation of the black graph.
a)
y
y x2
8
6
4
2
x
-4
-2
b)
0
2
-2
y x2
4
x
-2
0
-2
2
4
-4
-6
Lesson 5.2
3. Determine the values of h and k for each of the
following transformations. Write the equation in
the form y = (x - h)2 + k. Sketch the graph.
a) The parabola moves 3 units down and
2 units right.
b) The parabola moves 4 units left and
6 units up.
274
5. Describe the sequence of transformations that
you would apply to the graph of y = x 2 to
sketch each quadratic relation.
a) y = - 3(x - 1)2
1
b) y = (x + 3)2 - 8
2
c) y = 4(x - 2)2 - 5
2
d) y = x 2 - 1
3
question 5 on a separate grid. Use the properties
of the parabola and some additional points.
7. For each quadratic relation,
2
-4
Lesson 5.3
6. Sketch a graph of each quadratic relation in
y
6
4
entered as equations
of the form
y = (x - h)2 + k.
For each tick mark,
the scale on both
axes is 1. Determine as many of the equations
as you can.
Mid-Chapter Review
i) state the stretch/compression factor and the
horizontal/vertical translations
ii) determine whether the graph is reflected in
the x-axis
iii) state the vertex and the equation of the axis
of symmetry
iv) sketch the graph by applying transformations
to the graph of y = x 2
a) y = (x - 2)2 + 1
1
b) y = - (x + 4)2
2
c) y = 2(x + 1)2 - 8
d) y = - 0.25x 2 + 5
8. A parabola lies in only two quadrants. What
does this tell you about the values of a, h, and k?
Explain your thinking, and provide the equation
of a parabola as an example.
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Quadratic Models Using
Vertex Form
YOU WILL NEED
GOAL
• grid paper
• ruler
• graphing calculator
• spreadsheet program (optional)
Bakery Profits from Bread Sales
y
(1.75, 400)
400
Write the equation of the graph of a quadratic relation
in vertex form.
LEARN ABOUT the Math
The Best Bread Bakery wants to
determine its daily profit from bread
sales. This graph shows the data gathered
by the company.
Profit ($)
350
(0.75, 300)
300
250
200
?
What equation represents the relationship between the
price of bread and the daily profit from bread sales?
150
0
EXAMPLE 1
x
0.5
1 1.5 2 2.5
Price per loaf ($)
3
Connecting a parabola to the vertex form of its equation
Determine the equation of this quadratic relation from its graph.
Sabrina’s Solution
y = a(x - h)2 + k
Since the graph is a parabola and the coordinates of
the vertex are given, I decided to use vertex form.
y = a(x - 1.75)2 + 400
Since (1.75, 400) is the vertex, h 1.75 and
k 400. I substituted these values into the equation.
300 = a(0.75 - 1.75)2 + 400
300 = a(–1)2 + 400
300 = a + 400
–100 = a
The equation that represents the relationship
is y = –100(x - 1.75)2 + 400.
To determine the value of a, I chose the point
(0.75, 300) on the graph. I substituted these
coordinates for x and y in the equation.
I followed the order of operations and solved for the
value of a.
Reflecting
A.
What information do you need from the graph of a quadratic relation
to determine the equation of the relation in vertex form?
B.
You have used the standard, factored, and vertex forms of a quadratic
relation. Which form do you think is most useful for determining
the equation of a parabola from its graph? Explain why.
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APPLY the Math
EXAMPLE 2
Connecting information about a parabola to its equation
The graph of y = x 2 was stretched by a factor of 2 and reflected in the x-axis.
The graph was then translated to a position where its vertex is not visible in
the viewing window of a graphing calculator. Determine the quadratic
relation in vertex form from the partial graph displayed in the screen shot.
For each tick mark, the scale on the y-axis is 5, and the scale on the x-axis is 2.
18
4
Terri’s Solution
The graph was stretched by a factor of 2 and
reflected in the x-axis.
a = -2
y = - 2(x - h)2 + k
I substituted the value of a into the vertex form
of the quadratic relation.
The zeros of the graph are 3 and 13.
3 + 13
h =
2
h = 8
I determined the mean of the two zeros to calculate
the value of h. The vertex lies on the axis of symmetry,
which is halfway between the zeros of the graph.
18 = - 2(4 - 8)2 + k
18 = - 2(16) + k
18 = - 32 + k
50 = k
I saw that (4, 18) is a point on the graph. By
substituting these coordinates, as well as the value
I determined for h, I was able to solve for k.
The equation of the graph is y = - 2(x - 8)2 + 50.
EXAMPLE 3
Selecting a strategy to determine a quadratic model
The amount of gasoline that a car consumes depends on its speed. A group
of students decided to research the relationship between speed and fuel
consumption for a particular car. They collected the data in the table.
Determine an equation that models the relationship between speed and fuel
consumption.
Speed (km/h)
10
20
30
40
50
60
70
80
90
Gas Consumed
(litres/100 km)
9.2
8.1
7.4
7.2
6.4
6.1
5.9
5.8
6.0
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100
6.3
110
7.5
120
8.4
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Gas (litres/100 km)
Eric’s Solution: Representing a relation with a scatter plot and determining
the equation algebraically
10
y
I constructed a scatter plot to display the data and
drew a curve of good fit. Since the curve looked
parabolic and I knew that I could estimate the
coordinates of the vertex. I estimated the
coordinates of the vertex to be about (75, 5.8).
-8
6
4
vertex
(75, 5.8)
2
x
0
20 40 60 80 100 120 140
Speed (km/h)
y = a(x - h)2 + k
y = a(x - 75)2 + 5.8
I decided to use the vertex form of the equation.
I substituted the estimated values (75, 5.8) into the
general equation.
6.0 = a(90 - 75)2 + 5.8
From the table, I knew that the point (90, 6.0) is
close to the curve. I substituted the coordinates
of this point for x and y to determine a.
6.0 = a(15)2 + 5.8
6.0 = 225a + 5.8
0.2 = 225a
0.0009 ⬟ a
I solved for a.
The equation that models the data is
y = 0.0009(x - 75)2 + 5.8.
Gas Consumption
Gas (L/100 km)
10.0
6.0
y 0.0009x2 0.1317x
10.534
4.0
2.0
0.0
NEL
I checked my equation using a spreadsheet.
I entered the data from the table. I used column A
for the Speed values and column B for the Gas
values. I created a graph, added a trend line using
quadratic regression of order 2, and chose the
option to display the equation on the graph.
8.0
Tech
0
50
100
Speed (km/h)
150
Support
For help creating a scatter plot
and performing a regression
analysis using a spreadsheet,
see Appendix B-35.
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y = 0.0009(x - 75)2 + 5.8
y = 0.0009(x 2 - 150x + 5625) + 5.8
y = 0.0009x 2 - 0.135x + 5.0625 + 5.8
y = 0.0009x 2 - 0.135x + 10.8625
The spreadsheet equation was in standard form,
but my equation was in vertex form. To compare
the two equations, I expanded my equation.
The two equations are very close, so they are both
good quadratic models for this set of data.
Gillian’s Solution: Selecting a graphing calculator and an informal curve-fitting process
I entered the data into L1 and L2 in the data editor
of a graphing calculator and created a scatter plot.
y = a(x - 75)2 + 5.8
The points had a parabolic pattern, so I estimated
the coordinates of the vertex to be about (75, 5.8).
I substituted these coordinates into the general
equation.
Since the parabola opens upward, I knew that
a 7 0. I used a = 1 and entered the equation
y = 1(x - 75)2 + 5.8 into Y1 of the equation
editor. Then I graphed the equation. The location
of the vertex looked good, but the parabola wasn’t
wide enough.
I decreased the value of a to a = 0.1, but the
parabola still wasn’t wide enough.
I decreased the value of a several more times until
I got a good fit. I found that a = 0.0009 worked
fairly well.
Tech
An equation that models the relationship between speed
and fuel consumption is y = 0.0009(x - 75)2 + 5.8.
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5.4 Quadratic Models Using Vertex Form
Support
For help creating a scatter plot
using a TI-83/84 graphing
calculator, see Appendix B-10.
If you are using a TI-nspire, see
Appendix B-46.
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5.4
I checked my equation by comparing it with the
equation produced by quadratic regression on the
graphing calculator. To do this, I had to expand
my equation.
y = 0.0009(x - 75)2 + 5.8
y = 0.0009(x 2 - 150x + 5625) + 5.8
y = 0.0009x 2 - 0.135x + 5.0625 + 5.8
y = 0.0009x 2 - 0.135x + 10.8625
The two equations are close, so they are both
good models for this set of data.
Tech
Support
For help performing a
quadratic regression analysis
using a TI-83/84 graphing
calculator, see Appendix B-10.
If you are using a TI-nspire,
see Appendix B-46.
In Summary
Key Idea
• If you know the coordinates of the vertex (h, k) and one other point
on a parabola, you can determine the equation of the relation using
y = a(x - h)2 + k.
Need to Know
• To determine the value of a, substitute the coordinates of a point
on the graph into the general equation and solve for a:
• If (h, k) = (., .), then y = a(x - .)2 + ..
• If a point on the graph has coordinates x = . and y = ., then, by
substitution, . = a(. - .)2 + ..
• Since a is the only remaining unknown, its value can be determined
by solving the equation.
• The vertex form of an equation
y
(h, k)
can be determined using the
zeros of the graph. The axis of
symmetry is x = h, where h is
the mean of the zeros.
• You can convert a quadratic
x
(r, 0)
(s, 0)
equation from vertex form to
standard form by expanding
and then collecting like terms.
r s
x —
2 h
NEL
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CHECK Your Understanding
1. Match each equation with the correct graph.
a) y = 2x 2 – 8
b) y = (x + 3)2
i)
c) y = - 2(x – 4)2 + 8
d) y = (x – 3)2 – 8
iii)
y
10
2
8
x
6
-4
4
x
0
2
-2
4
6
2
-2
-8
-10
iv)
y
4
8
x
0
-2
y
10
2
-2
4
-6
8
-4
ii)
-2
0
-4
2
-2
y
4
2
4
6
6
8
4
2
-4
-6
x
-8 -6 -4
-2
0
-8
-2
-10
-4
2
2. The vertex of a quadratic relation is (4, 12).
a) Write an equation to describe all parabolas with this vertex.
b) A parabola with the given vertex passes through point (13, 15).
Determine the value of a for this parabola.
c) Write the equation of the relation for part b).
d) State the transformations that must be applied to y x2 to obtain
the quadratic relation you wrote for part c).
e) Graph the quadratic relation you wrote for part c).
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5.4 Quadratic Models Using Vertex Form
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5.4
PRACTISING
3. Write the equation of each parabola in vertex form.
a)
c)
y
6
4
2
2
2
(2, 1) x
-2
b)
0
2
-2
-4
-2
d)
y
6
(2, 0)
4
(–2, 2)
-2
4
-4
0
2
4
0
-4 -2
(–1, –2)
-2
f)
y
x
2
-2
2
-2
2
(3, 1)
x
2
x
-4
0
4
4
y
6
4
4
-4
e)
y
6
4
-4
0
4
y
4
2
(–3, 2)
-4
2
-2
x
x
-2
0
-2
2
4
-4
4. The following transformations are applied to the graph of y = x 2.
Determine the equation of each new relation.
a) a vertical stretch by a factor of 4
b) a translation of 3 units left
c) a reflection in the x-axis, followed by a translation 2 units up
1
d) a vertical compression by a factor of
2
e) a translation of 5 units right and 4 units down
f ) a vertical stretch by a factor of 2, followed by a reflection
in the x-axis and a translation 1 unit left
5. Write the equation of a parabola with each set of properties.
a) vertex at (0, 4), opens upward, the same shape as y x2
b) vertex at (5, 0), opens downward, the same shape as y x2
c) vertex at (2, 3), opens upward, narrower than y x2
d) vertex at (3, 5), opens downward, wider than y x2
e) axis of symmetry x 4, opens upward, two zeros,
narrower than y x2
f ) vertex at (3, – 4), no zeros, wider than y x2
6. Determine the equation of a quadratic relation in vertex form, given
K
NEL
the following information.
a) vertex at (2, 3), passes through (4, 1)
b) vertex at (1, 1), passes through (0, 1)
c) vertex at (2, 3), passes through (5, 6)
d) vertex at (2, 5), passes through (1, 4)
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7. Each table of values defines a parabola. Determine the equation of the
axis of symmetry of the parabola, and write the equation in vertex form.
a)
x
y
2
b)
x
y
33
0
12
3
13
1
4
4
1
2
4
5
3
3
12
6
1
4
28
8. A child kicks a soccer ball so that it barely clears a 2 m fence. The
soccer ball lands 3 m from the fence. Determine the equation, in
vertex form, of a quadratic relation that models the path of the ball.
9. Data for DVD sales in Canada, over several years, are given in the table.
Year
2002
2003
2004
2005
2006
x, Years Since 2002
0
1
2
3
4
DVDs Sold (1000s)
1446
3697
4573
4228
3702
a) Using graphing technology, create a scatter plot to display the data.
b) Estimate the vertex of the graph you created for part a). Then
determine an equation in vertex form to model the data.
c) How many DVDs would you expect to be sold in 2010?
d) Check the accuracy of your model using quadratic regression.
10. A school custodian finds a tennis ball on the roof of the school and
A
throws it to the ground below. The table gives the height of the ball
above the ground as it moves through the air.
Time (s)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Height (m) 5.00
11.25
15.00
16.25
15.00
11.25
5.00
a) Do the data appear to be linear or quadratic? Explain.
b) Create a scatter plot, and draw a quadratic curve of good fit.
c) Estimate the coordinates of the vertex.
d) Determine an algebraic relation in vertex form to model the data.
e) Use your model to predict the height of the ball at 2.75 s and 1.25 s.
f ) How effective is your model for time values that are greater than
3.5 s? Explain.
g) Check the accuracy of your model using quadratic regression.
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11. A chain of ice cream stores sells $840 of ice cream cones per day. Each
ice cream cone costs $3.50. Market research shows the following trend
in revenue as the price of an ice cream cone is reduced.
Price ($)
3.50
3.00
2.50
2.00
1.50
1.00
0.50
Revenue ($)
840
2520
3600
4080
3960
3240
1920
a) Create a scatter plot, and draw a quadratic curve of good fit.
b) Determine an equation in vertex form to model this relation.
c) Use your model to predict the revenue if the price of an ice cream
cone is reduced to $2.25.
d) To maximize revenue, what should an ice cream cone cost?
e) Check the accuracy of your model using quadratic regression.
12. This table shows the number of imported cars that were sold
in Newfoundland between 2003 and 2007.
Year
2003
2004
2005
2006
2007
Sales of Imported
Cars (number sold)
3996
3906
3762
3788
4151
a) Create a scatter plot, and draw a quadratic curve of good fit.
b) Determine an algebraic equation in vertex form to model
this relation.
c) Use your model to predict how many imported cars were sold
in 2008.
d) What does your model predict for 2006? Is this prediction
accurate? Explain.
e) Check the accuracy of your model using quadratic regression.
13. The Lion’s Gate Bridge in Vancouver, British Columbia, is a
T
suspension bridge that spans a distance of 1516 m. Large cables are
attached to the tops of the towers, 50 m above the road. The road is
suspended from the large cables by many smaller vertical cables. The
smallest vertical cable measures about 2 m. Use this information to
determine a quadratic model for the large cables.
14. A model rocket is launched from the ground. After 20 s, the rocket
C
NEL
reaches a maximum height of 2000 m. It lands on the ground after
40 s. Explain how you could determine the equation of the relationship
between the height of the rocket and time using two different strategies.
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15. The owner of a small clothing company wants to create a
mathematical model for the company’s daily profit, p, in dollars, based
on the selling price, d, in dollars, of the dresses made. The owner has
noticed that the maximum daily profit the company has made is
$1600. This occurred when the dresses were sold for $75 each. The
owner also noticed that selling the dresses for $50 resulted in a profit
of $1225. Using a quadratic relation to model this problem, create an
equation for the company’s daily profit.
16. Compare the three forms of the equation of a quadratic relation using
this concept circle. Under what conditions would you use one form
instead of the other forms when trying to connect a graph to its
equation? Explain your thinking.
Quadratic Relations
Factored
Form
Standard
Form
Vertex
Form
Extending
17. The following transformations are applied to a parabola with the
equation y = 2(x + 3)2 – 1. Determine the equation that will result
after each transformation.
a) a translation 4 units right
b) a reflection in the x-axis
c) a reflection in the x-axis, followed by a translation 5 units down
d) a stretch by a factor of 6
1
e) a compression by a factor of , followed by a reflection in the y-axis
4
18. The vertex of the parabola y = 3x 2 + bx + c is at (1, 4).
Determine the values of b and c.
19. Determine an algebraic expression for the solution, x, to the equation
0 = a(x - h)2 + k. Do not expand the equation.
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5.4 Quadratic Models Using Vertex Form
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Solving Problems Using
Quadratic Relations
YOU WILL NEED
GOAL
Model and solve problems using the vertex form of a quadratic
relation.
• grid paper
• ruler
LEARN ABOUT the Math
Smoke jumpers are firefighters who parachute into remote locations to
suppress forest fires. They are often the first people to arrive at a fire. When
smoke jumpers exit an airplane, they are in free fall until their parachutes open.
A quadratic relation can be used to determine the height, H, in metres, of
a jumper t seconds after exiting an airplane. In this relation, a = - 0.5g,
where g is the acceleration due to gravity. On Earth, g = 9.8 m/s2.
?
If a jumper exits an airplane at a height of 554 m, how long will
the jumper be in free fall before the parachute opens at 300 m?
EXAMPLE 1
Connecting information from a problem
to a quadratic model
a) Determine the quadratic relation that will model the height, H, of the
smoke jumper at time t.
b) Determine the length of time that the jumper is in free fall.
Conor’s Solution
a) H = a(t - h)2 + k
NEL
I decided to use the vertex form
of the quadratic relation because
the problem contains
information about the vertex.
Environment Connection
In a recent year, 3596 of the
7290 forest fires in Canada
were caused by human
activities such as careless
smoking, campfires, use of
welding equipment, or
operation of a motor vehicle.
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H = a(t - 0)2 + 554
H = - 0.5(9.8)(t - 0)2 + 554
- 4.9(t - 0)2 + 554 is an
Since a 0.5g and
g 9.8 m/s2, I substituted these
values into the vertex form of the
equation.
H = - 4.9t 2 + 554 is an equation
in standard form for the quadratic
relation that models this situation.
I noticed that the value of a is the
same in both vertex form and
standard form. This makes sense
because the parabolas would not
be congruent if they were different.
H =
equation in vertex form for the
quadratic relation that models
this situation.
b)
The vertex is the point at which
the jumper exited the plane. So
the vertex has coordinates (0, 554).
I substituted these coordinates into
the general equation.
300 = - 4.9t 2 + 554
-254 = - 4.9t 2
-254
-4.9t 2
=
-4.9
-4.9
2
51.84 = t
151.84 = t
7.2 ⬟ t, since t 7 0
The jumper is in free fall
for about 7.2 s.
Because the parachute opened
at 300 m, I substituted 300 for H.
Then I solved for t.
In this situation, time can’t be
negative. So, I didn’t use the
negative square root of 51.84.
Reflecting
286
A.
Why was zero used for the t-coordinate of the vertex?
B.
How would the equation change if the jumper hesitated for 2 s before
exiting the airplane, after being given the command to jump?
C.
Why was the vertex form easier to use than either of the other two
forms of a quadratic relation in this problem?
5.5 Solving Problems Using Quadratic Relations
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APPLY the Math
EXAMPLE 2
Solving a problem using a quadratic model
The underside of a concrete railway underpass forms a parabolic arch. The
arch is 30.0 m wide at the base and 10.8 m high in the centre. The upper
surface of the underpass is 40.0 m wide. The concrete is 2.0 m thick at the
centre. Can a truck that is 5 m wide and 7.5 m tall get through this underpass?
Lisa’s Solution
y
4
2 (0, 0)
0
-20 -16 -12 -8 -4
(–15, –10.8)
-4
-6
-8
-10
-12
-14
4
x
8
12 16 20
(15, –10.8)
The vertex of the parabola is at the origin, so I did
not translate y = ax 2.
y = ax 2
- 10.8 = a(15)2
-10.8 = 225a
- 0.048 = a
y = - 0.048x 2 is the quadratic relation that models
the arch of the railway underpass.
y
4
2 (0, 0)
-20 -16 -12 -8 -4
(–15, –10.8)
I started by drawing a diagram. I used a grid and
marked the top of the arch as (0, 0). The upper
surface of the underpass is 2 m above the top of
the arch at the centre. The arch is 10.8 m high in
the centre and 30 m wide at the base (or 15 m
wide on each side). I marked the points
(15, 10.8) and (15, 10.8) and drew a
parabola through these two points and the origin.
-4
-6
-8
-10
-12
-14
0
4
x
8
12 16 20
(15, –10.8)
I determined the value of a by substituting the
coordinates of a point on the graph, (15, 10.8),
into this equation. Then I solved for a.
The truck has the best chance of getting through
the underpass if it passes through the centre. Since
the truck is 5 m wide, this means that the position
of the right corner of the truck has an x-coordinate
of 2.5. I substituted x = 2.5 into the equation to
check the height of the underpass at this point.
y = - 0.048(2.5)2
y = - 0.048(6.25)
y = - 0.3
Height at (2.5, - 0.3) = 10.8 - 0.3
= 10.5
I determined the height from the ground at this
point by subtracting 0.3 from 10.8.
The truck can get through. Since the truck is
7.5 m tall, there is 3 m of clearance.
The truck can get through the underpass, even if it
is a little off the centre of the underpass.
NEL
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EXAMPLE 3
Selecting a strategy to determine
the vertex form
Write the quadratic relation y = x 2 - 4x - 5 in vertex form, and sketch
the graph by hand.
Coral’s Solution
y = x 2 - 4x - 5
y = (x + 1)(x - 5)
I rewrote the equation of the
quadratic relation in factored
form because I knew that I could
determine the coordinates of the
vertex from this form.
Zeros:
0 = (x + 1)(x - 5)
x = - 1 and x = 5
The axis of symmetry is
-1 + 5
so x 2.
x =
2
I set y 0 to determine the zeros.
I used the zeros to determine the
equation of the axis of symmetry.
y = (2)2 - 4(2) - 5
y = 4 - 8 - 5
y = -9
I substituted x = 2 into the
standard form of the equation to
solve for y.
The vertex is at (h, k) = (2, -9).
The coefficient of x 2 is a = 1.
I knew that the value of a must
be the same in the standard,
factored, and vertex forms. If it
were different, the parabola
would have different widths.
The relation is y = (x - 2)2 - 9
in vertex form.
I substituted what I knew into the
vertex form, y = a (x - h)2 + k.
y
16
y x2
12
8
4
x
-6
-4
-2
0
-4
2
4
6
8
I sketched the graph of y = x 2
and translated each point 2 units
right and 9 units down.
-8
-12
y (x 2)2 9
-16
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EXAMPLE 4
Representing a situation with a quadratic
model
The Next Cup coffee shop sells a special blend of coffee for $2.60 per mug.
The shop sells about 200 mugs per day. Customer surveys show that for
every $0.05 decrease in the price, the shop will sell 10 more mugs per day.
a) Determine the maximum daily revenue from coffee sales and the price
per mug for this revenue.
b) Write an equation in both standard form and vertex form to model
this problem. Then sketch the graph.
Dave’s Solution: Connecting the zeros of a parabola
to the vertex form of the equation
a) Let x represent the number of
$0.05 decreases in price, where
Revenue (price)(mugs sold).
r = (2.60 - 0.05x)(200 + 10x)
I defined a variable that connects
the price per mug to the number
of mugs sold.
I used the information in the
problem to write expressions for
the price per mug and the
number of mugs sold in terms
of x. If I drop the price by $0.05,
x times, then the price per mug
is 2.60 0.05x and the number
of mugs sold is 200 10x.
I used my expressions to write a
relationship for daily revenue, r.
0 = (2.60 - 0.05x)(200 + 10x)
2.60 - 0.05x = 0, so x 52
or
200 + 10x = 0, so x 20
Since the equation is in factored
form, the zeros of the equation
can be calculated by letting
r 0 and solving for x.
52 + (- 20)
2
x = 16
I used the zeros to determine
the equation of the axis of
symmetry.
x =
r = [2.60 - 0.05(16)][200 + 10(16)]
r = (1.80)(360)
r = 648
The maximum daily revenue
is $648.
NEL
The maximum value occurs
at the vertex. To calculate it,
I substituted the x-value for
the axis of symmetry into the
revenue equation.
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Price per mug for maximum revenue
= 2.60 - 0.05(16)
= 1.80
The coffee shop should sell each
mug of coffee for $1.80 to achieve
a maximum daily revenue of $648.
I substituted x 16 into the
expression for the price
per mug.
The maximum value occurs
at the vertex of a quadratic
relation. To write the equation
in vertex form, substitute
the values of h and k into the
vertex form of the general
equation.
b) In this relation, the maximum
value is r = 648. It occurs when
x = 16.
The vertex is (16, 648) (h, k).
r = a(x - 16)2 + 648
When x = 0,
r = (2.60)(200) = 520
520 = a(0 - 16)2 + 648
520 = a( -16)2 + 648
-128 = 256a
-0.5 = a
The equation in vertex form is
r = - 0.5(x - 16)2 + 648.
Since the coffee shop sells 200
mugs of coffee when the price
is $2.60 per mug, the point
(0, 520) is on the graph. I
substituted these coordinates
into the equation and solved
for a.
r = - 0.5(x - 16)2 + 648
r = - 0.5(x 2 - 32x + 256) + 648
r = - 0.5x 2 + 16x - 128 + 648
r = - 0.5x 2 + 16x + 520
I expanded to get the equation
in standard form.
The equation in standard form is
r = - 0.5x 2 + 16x + 520.
r
700
(16, 648)
600
The vertex is at (16, 648).
The zeros are at (52, 0) and
(20, 0).
The y-intercept is at (0, 520).
500
400
300
I used these points to sketch
the graph of the relation.
200
100
x
-20 -10
290
0
10 20 30 40 50
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Toni’s Solution: Selecting a graphing calculator to determine
the quadratic model
a)
Once I created the revenue
equation, I entered it into the
equation editor as Y1.
I graphed the revenue equation.
I had to adjust the window
settings until I could see the
zeros and the vertex.
Since the vertex in this model
represents the maximum value,
I determined it using the
maximum operation.
The vertex is at (16, 648).
The maximum daily revenue
is $648.
Selling price 2.60 0.05(16)
1.80
Each mug of coffee should be
sold for $1.80 to maximize the
daily revenue.
b) The equation in standard form is
y = (2.60 – 0.05x)(200 + 10x)
y = - 0.5x 2 + 16x + 520
a = - 0.5
The vertex is at (16, 648) (h, k).
y = a(x - h)2 + k
y = - 0.5(x - 16)2 + 648
This is the equation in vertex form.
NEL
Tech
Support
For help determining the
maximum value of a relation
using a TI-83/84 graphing
calculator, see Appendix B-9.
If you are using a TI-nspire,
see Appendix B-45.
The maximum value is y = 648.
This means that the maximum
daily revenue is $648. It occurs
when x = 16.
Since the calculator has already
produced the graph of the
model, I only needed to
determine the vertex form. I
took the revenue equation and
expanded it to get the equation
in standard form.
Since the value of a is the same
in all forms of a quadratic
relation, I used it along with the
coordinates of the vertex, and
substituted to obtain the
equation in vertex form.
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In Summary
Key Idea
• All quadratic relations can be expressed in vertex form and standard
form. Quadratic relations that have zeros can also be expressed
in factored form.
• For any parabola, the value of a is the same in all three forms
of the equation of the quadratic relation.
Need to Know
• The y-coordinate of the vertex of a parabola represents the maximum or
minimum value of the quadratic relation. The coordinates of the vertex
are easily determined from the vertex form of the equation.
• If a situation can be modelled by a quadratic relation of the form
y a(x h)2 k, the maximum or minimum value of y is k and
it occurs when x h.
xh
a 0, k is a maximum
y
(h, k)
x
(h, k)
a 0, k is a minimum
xh
• If y ax2 bx c can be factored as a product of first-degree
binomials and a constant, y a(x r)(x s), then this equation can be
used to determine the vertex form of the quadratic relation as follows:
r + s
• Use x to determine the equation of the axis of symmetry.
2
This gives you the value of h.
292
r + s
•
Substitute x into y ax2 bx c to determine the
2
y-coordinate of the vertex. This gives you the value of k.
•
Substitute the values of a, h, and k into y a(x h)2 k.
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5.5
CHECK Your Understanding
1. Use the given information to determine the equation of each quadratic
relation in vertex form, y = a(x - h)2 + k .
a) a = 2, vertex at (0, 3)
c) a = - 1, vertex at (3, 2)
b) a = - 3, vertex at (2, 0)
d) a = 0.5, vertex at (3.5, 18.3)
2. Determine each maximum or minimum value in question 1.
3. The arch of the bridge in this photograph can be modelled by a parabola.
a) Determine an equation of the parabola.
b) On the upper part of the bridge, three congruent arches are visible
in the first and second quadrants. What can you conclude about
the value of a in the equations of the parabolas that model these
arches? Explain.
4
y
2
x
-5
0
5
-2
-4
PRACTISING
4. Determine the equation of a quadratic relation in vertex form,
given the following information.
a) vertex at (0, 3), passes through (2, 5)
b) vertex at (2, 0), passes through (5, 9)
c) vertex at (3, 2), passes through (1, 14)
d) vertex at (5, 3), passes through (1, 8)
5. Determine the equation of each parabola in vertex form.
a)
c)
y
6
4
2
(1, 0)
-2
0
(2, 0)
4
2
-2
x
-2
2
-2
(6, 0) x
4
8
6
-6
y
6
0
-4
(–1, –4)
-6
b)
(4, 4)
4
2
-4
y
6
(6, 6)
d)
y
6
(0, 4)
4
4
(2, 2)
2
2
x
x
-2
NEL
0
-2
2
4
(4, –2)
6
8
-4
-2
0
-2
-4
-4
-6
-6
2
4
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6. Write each equation in question 5 in standard form and factored form.
7. A quadratic relation has zeros at –2 and 8, and a y-intercept of 8.
K
Determine the equation of the relation in vertex form.
8. The quadratic relation y = 2(x + 4)2 - 7 is translated 5 units right
and 3 units down. What is the minimum value of the new relation?
Write the equation of this relation in vertex form.
9. Express each equation in standard form and factored form.
a) y = (x - 4)2 - 1
b) y = 2(x + 1)2 - 18
c) y = –(x + 5)2 + 1
d) y = –3(x + 3)2 + 75
10. Express each equation in factored form and vertex form.
a) y = 2x 2 - 12x
b) y = –2x 2 + 24x - 64
c) y = 2x 2 - x - 6
d) y = 4x 2 + 20x + 25
11. A dance club has a $5 cover charge and averages 300 customers
on Friday nights. Over the past several months, the club has changed
the cover price several times to see how this affects the number
of customers. For every increase of $0.50 in the cover charge,
the number of customers decreases by 30. Use an algebraic model
to determine the cover charge that maximizes revenue.
12. The graph of y = - 2(x + 5)2 + 8 is translated so that its new zeros
are 4 and 2. Determine the translation that was applied to the
original graph.
13. The average ticket price at a regular movie theatre (all ages) from 1995
to 1999 can be modelled by C = 0.06t 2 - 0.27t + 5.36, where C
is the price in dollars and t is the number of years since 1995 (t = 0
for 1995, t = 1 for 1996, and so on).
a) When were ticket prices the lowest during this period?
b) What was the average ticket price in 1998?
c) What does the model predict the average ticket price will be
in 2010?
d) Write the equation for the model in vertex form.
14. A bridge is going to be constructed over a river. The underside of the
A
bridge will form a parabolic arch, as shown in the picture. The river is
18 m wide and the arch will be anchored on the ground, 3 m back
from the riverbank on both sides. The maximum height of the arch
must be between 22 m and 26 m above the surface of the river. Create
two different equations to represent arches that satisfy these conditions.
Then use graphing technology to graph your equations on the same grid.
18 m
3m
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3m
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15. A movie theatre can accommodate a maximum of 350 moviegoers per
day. The theatre operators have been changing the admission price to
find out how price affects ticket sales and profit. Currently, they charge
$11 a person and sell about 300 tickets per day. After reviewing their
data, the theatre operators discovered that they could express the
relation between profit, P, and the number of $1 price increases, x,
as P = 20(15 - x)(11 + x).
a) Determine the vertex form of the profit equation.
b) What ticket price results in the maximum profit? What is the
maximum profit? About how many tickets will be sold at this price?
16. The underside of a bridge forms a parabolic arch. The arch has a
maximum height of 30 m and a width of 50 m. Can a sailboat pass
under the bridge, 8 m from the axis of symmetry, if the top of its mast
is 27 m above the water? Justify your solution.
17. A parabola has a y-intercept of 4 and passes through points (2, 8)
T and (1, 1). Determine the vertex of the parabola.
18. Serena claims that the standard form of a quadratic relation is best for
C
solving problems where you need to determine the maximum or
minimum value, and that the vertex form is best to use to determine a
parabola’s zeros. Do you agree or disagree? Explain.
Extending
19. The equation of a parabola is y = a(x - 1)2 + q , and the points
(1, 9) and (1, 1) lie on the parabola. Determine the maximum
value of y.
20. A rectangular swimming pool has a row of water fountains along
each of its two longer sides. The two rows of fountains are 10 m
apart. Each fountain sprays an identical parabolic-shaped stream
of water a total horizontal distance of 8 m toward the opposite
side. Looking from one end of the pool, the streams of water
from the two sides cross each other in the middle of the pool
at a height of 3 m.
a) Determine an equation that represents a stream of water from
the left side and another equation that represents a stream of
water from the right side. Graph both equations on the same
set of axes.
b) Determine the maximum height of the water.
NEL
3m
8m
10 m
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• graphing calculator
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Curious Math
The Golden Rectangle
The golden rectangle is considered one of the most pleasing shapes to the
human eye. It is often used in architectural design, and it can be seen in
many famous works of art. For example, the golden rectangle can be seen in
Leonardo Da Vinci’s Mona Lisa and in the Parthenon in Athens, Greece.
One of the properties of the golden rectangle is its dimensions. When it is
divided into a square and a smaller rectangle, the smaller rectangle is similar
to the original rectangle.
Golden Rectangle
1
smaller
rectangle
square
x1
1
x
The ratio of the longer side to the shorter side in a golden rectangle is
called the golden ratio.
If the length of the shorter side is 1 unit, and if x represents the length
x
of the longer side, then = x is also the value of the golden ratio. A
1
quadratic relation can be used to determine the value of the golden ratio.
1. Create a proportion statement to compare the golden ratio with the
ratio of the lengths of the corresponding sides in the smaller rectangle.
2. Substitute the values in the diagram into your proportion statement.
Then rearrange your proportion statement to obtain a quadratic
relation.
3. Using graphing technology, graph the quadratic relation that
corresponds to this equation.
4. What feature of the graph represents the value of the golden ratio?
5. Use graphing technology to determine the value of the golden ratio,
correct to three decimal places.
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Connecting Standard
and Vertex Forms
GOAL
Sketch or graph a quadratic relation with an equation of the form
y ax2 bx c using symmetry.
YOU WILL NEED
• grid paper
• ruler
INVESTIGATE the Math
Many places hold a fireworks display on Canada Day. Clayton,
a member of the local fire department, launches a series of rockets
from a barge that is floating in the middle of the lake. Each rocket
is choreographed to explode at the correct time. The equation
h = - 5t 2 + 40t + 2 can be used to model the height, h, of each rocket
in metres above the water at t seconds after its launch. A certain rocket
is scheduled to explode 3 min 21 s into the program.
Safety Connection
?
Assuming that the rocket will explode at its highest point, when
should Clayton launch it from the barge so it will explode at the
correct time?
A.
What information do you need to determine so that you can model
the height of the rocket?
B.
Copy and complete the table of values at the right for the rocket.
Then plot the points, and sketch the graph of this relation.
C.
What happens to the rocket between 8 s and 9 s after it is launched?
D.
The axis of symmetry of a quadratic relation can be determined from
the zeros. In this problem, however, there is only one zero because
t 7 0. Suggest another way to determine the axis of symmetry.
2
The rocket is 2 m above the water when it is launched. When will
the rocket be at the same height again? Write the coordinates of these
two points.
5
E.
F.
G.
NEL
Consider the coordinates of the two points for part E. Why must the
axis of symmetry be the same distance from both of these points?
What is the equation of the axis of symmetry?
Fireworks can cause serious
injury when handled
incorrectly. It is safer to watch
a community display than to
create your own.
Time (s)
Height (m)
0
2
1
37
3
4
6
7
8
9
How does knowing the equation of the axis of symmetry help you
determine the vertex of a parabola? What is the vertex of this parabola?
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When should Clayton launch the rocket to ensure that it explodes
3 min 21 s into the program?
Reflecting
I.
Can you determine the maximum height of the rocket directly from
the standard form of the quadratic relation h = - 5t 2 + 40t + 2 ?
Explain.
J.
How did you determine the vertex, even though one of the zeros
of the quadratic relation was unknown?
K.
Write the quadratic relation in vertex form. How can you compare this
equation with the equation given in standard form to determine
whether they are identical?
APPLY the Math
EXAMPLE 1
Connecting the vertex form to partial factors of the equation
Determine the maximum value of the quadratic relation y = - 3x 2 + 12x + 29.
Michelle’s Solution
y = - 3x 2 + 12x + 29
I tried to factor the expression, but I couldn’t
determine two integers with a product of
(3) 29 and a sum of 12. This means that I can’t
use the zeros to help me.
y = x(- 3x + 12) + 29
I had to determine the axis of symmetry, since the
vertex (where the maximum value occurs) lies on it.
To do this, I had to locate two points with the same
y-coordinate. I removed a partial factor of x from
the first two terms.
When y = 29,
29 = x( - 3x + 12) + 29
x = 0 or - 3x + 12 = 0
x = 0
x = 4
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5.6 Connecting Standard and Vertex Forms
I noticed that the y-value will be 29 if either factor
in the equation equals 0. I decided to determine
the two points on the parabola with a y-coordinate
of 29 by setting each partial factor equal to 0 and
solving for x.
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5.6
y
45
I knew that this would work because it was like
translating the graph down 29 units to make the
points with a y-coordinate of 29 turn into points
with a y-coordinate of 0.
40
35
y 29
30
25
This let me determine the zeros of the new
translated graph.
I noticed that the axis of symmetry was the same
for the two graphs.
20
15
10
5
x
-8 -6
-4
-2
0
2
-5
y 3x2 12x 29
4
6
8
y 3x2 12x
(0, 29) and (4, 29) are on the graph of the original
quadratic relation. This makes sense since the points
(0, 0) and (4, 0) are the zeros of the translated graph.
These points are the same distance from the axis of
symmetry. So, I know that the axis of symmetry is
halfway between x 0 and x 4.
The equation of the axis of symmetry is
0 + 4
x =
so x 2.
2
y = - 3(2)2 + 12(2) + 29
y = - 12 + 24 + 29
I calculated the mean of the x-coordinates of these
points to determine the axis of symmetry.
y = 41
The y-coordinate of the vertex is the maximum
value because the graph opens downward. To
determine the maximum, I substituted x 2 into
y 3x2 12x 29.
The maximum value is 41.
EXAMPLE 2
Selecting a partial factoring strategy to sketch the graph
of a quadratic relation
Express the quadratic relation y = 2x 2 + 8x + 5 in vertex form.
Then sketch a graph of the relation by hand.
Marnie’s Solution
y = 2x 2 + 8x + 5
y = x(2x + 8) + 5
This equation cannot be factored fully since you
can’t determine two integers with a product of
2 5 and a sum of 8. I removed a partial factor
of x from the first two terms.
x = 0 or 2x + 8 = 0
x = 0
x = -4
I found two points with a y-coordinate of 5 by setting
each partial factor equal to 0. Both of these points
are the same distance from the axis of symmetry.
The points (0, 5) and (4, 5) are on the parabola.
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The axis of symmetry is x = - 2 .
I found the axis of symmetry by calculating the
mean of the x-coordinates of these points.
At the vertex,
y = 2(- 2)2 + 8(- 2) + 5
y = -3
Since the parabola is symmetric, the vertex is on
the line x 2. I substituted this value into the
relation.
The vertex of the parabola is at (2, 3).
In vertex form, the equation of the parabola is
y = 2(x + 2)2 - 3.
y 2x2 8x 5
y
8
6
(–4, 5)
I know that the value of a is the same in standard
form and vertex form. In this case, a 2.
The parameter a is positive, so the parabola opens
upward.
(0, 5)
I used the vertex and the two points I found, (0, 5)
and (4, 5), to sketch the parabola.
4
2
x
-6 -4
-2
0
-2
2
4
(–2, –3) -4
In Summary
Key Idea
• If a quadratic relation is in standard form and cannot be factored fully,
you can use partial factoring to help you determine the axis of symmetry
of the parabola. Then you can use the axis of symmetry to determine
the coordinates of the vertex.
Need to Know
• If y ax2 bx c cannot be factored, you can express the relation
in the partially factored form y x(ax b) c. Then you can use this
form to determine the vertex form:
• Set x(ax b) 0 and solve for x to determine two points on the
parabola that are the same distance from the axis of symmetry. Both
of these points have y-coordinate c.
• Determine the axis of symmetry, x h, by calculating the mean
of the x-coordinates of these points.
• Substitute x h into the relation to determine k, the y-coordinate
of the vertex.
• Substitute the values of a, h, and k into y a(x h)2 k.
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CHECK Your Understanding
1. Determine the equation of the axis of symmetry of a parabola that
passes through points (2, 8) and (6, 8).
2. Determine two points that are the same distance from the axis
of symmetry of the quadratic relation y = 4x 2 - 12x + 5.
3. Use partial factoring to determine the vertex form of the quadratic
relation y = 2x 2 - 10x + 11.
PRACTISING
4. A parabola passes through points (3, 0), (7, 0), and (9, 24).
a) Determine the equation of the axis of symmetry.
b) Determine the coordinates of the vertex, and write the equation
in vertex form.
c) Write the equation in standard form.
5. For each quadratic relation,
i) determine the coordinates of two points on the graph that are
the same distance from the axis of symmetry
ii) determine the equation of the axis of symmetry
iii) determine the coordinates of the vertex
iv) write the relation in vertex form
a) y = (x - 1)(x + 7)
b) y = x(x - 6) - 8
c) y = –2(x + 3)(x - 7)
d) y = x(3x + 12) + 2
e) y = x 2 + 5x
f ) y = x 2 - 11x + 21
6. The equation of one of these parabolas at the right is y = x 2 - 8x + 18.
K
Determine the equation of the other in vertex form.
4
7. For each quadratic relation,
i) use partial factoring to determine two points that are the same
distance from the axis of symmetry
ii) determine the coordinates of the vertex
iii) express the relation in vertex form
iv) sketch the graph
a) y = x 2 - 6x + 5
b) y = x 2 - 4x - 11
2
c) y = - 2x + 12x - 11
y
6
2
x
-2
0
-2
2
4
6
d) y = - x 2 - 6x - 13
1
e) y = - x 2 + 2x - 3
2
f ) y = 2x 2 - 10x + 11
8. Use two different strategies to determine the equation of the axis
C
NEL
of symmetry of the parabola defined by y = - 2x 2 + 16x - 24.
Which strategy do you prefer? Explain why.
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9. Determine the values of a and b in the relation y = ax 2 + bx + 7
T if the vertex is located at (4, 5).
10. Determine the values of a and b in the relation y = ax 2 + bx + 8
if the vertex is located at (1, 7).
11. A model rocket is launched straight up, with an initial velocity
of 150 m/s. The height of the rocket can be modelled by
h = - 5t 2 + 150t, where h is the height in metres and t is the elapsed
time in seconds. What is the maximum height reached by the rocket?
12. A baseball is hit from a height of 1 m. The height, h, of the ball in
A
metres after t seconds can be modelled by h = - 5t 2 + 9t + 1.
Determine the maximum height reached by the ball.
13. A movie theatre can accommodate a maximum of 450 moviegoers per
day. The theatre operators have determined that the profit per day, P, is
related to the ticket price, t, by P = - 30t 2 + 450t - 790. What
ticket price will maximize the daily profit?
14. The world production of gold from 1970 to 1990 can be modelled by
G = 1492 - 76t + 5.2t 2, where G is the number of tonnes of gold
and t is the number of years since 1970 (t = 0 for 1970, t = 1 for
1971, and so on).
a) During this period, when was the minimum amount of gold mined?
b) What was the least amount of gold mined in one year?
c) How much gold was mined in 1985?
15. Create a concept web that summarizes the different algebraic strategies
you can use to determine the axis of symmetry and the vertex of a
quadratic relation given in the form y = ax 2 + bx + c .
Extending
16. A farmer has $3000 to spend on fencing for two adjoining rectangular
pastures, both with the same dimensions. A local contracting company
can build the fence for $5.00/m. What is the largest total area that
the farmer can have fenced for this price?
17. A city transit system carries an average of 9450 people per day on its
buses, at a fare of $1.75 each. The city wants to maximize the transit
system’s revenue by increasing the fare. A survey shows that the
number of riders will decrease by 210 for every $0.05 increase in the
fare. What fare will result in the greatest revenue? How many daily
riders will they lose at this new fare?
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Chapter Review
FREQUENTLY ASKED Questions
Study
Q:
• See Lesson 5.4,
What information do I need about the graph of a quadratic
relation to write its equation in vertex form?
A:
You can write the equation in vertex form if you know the coordinates
of the vertex and one additional point on the graph.
Q:
What kinds of problems can you solve using the vertex
form of a quadratic relation?
A:
The vertex form of a quadratic relation is usually the easiest form to
use when you need to determine the equation of a parabola of good
fit on a scatter plot. The vertex form is also useful when you need to
determine the maximum or minimum value of a quadratic relation.
Q:
How can you relate the standard form of a quadratic
relation to its vertex form?
A:
The standard form of a quadratic relation, y = ax 2 + bx + c, can
be rewritten in vertex form if you know the value of a and the
coordinates of the vertex:
• If the quadratic relation can be written in factored form, you can
determine the zeros by setting each factor equal to zero. Calculating
the mean of the x-coordinates of the zeros gives you the axis of
symmetry and the x-coordinate of the vertex. Substitute the
x-coordinate of the vertex into the quadratic relation to determine
the y-coordinate of the vertex.
• If y = ax 2 + bx + c cannot be factored, you can use partial
factoring to express the equation in the form y = x(ax + b) + c.
Solving x(ax + b) = 0 gives two points with the same
y-coordinate, c. Calculating the mean of the x-coordinates of these
points gives you the axis of symmetry and the x-coordinate of the
vertex. Substitute the x-coordinate of the vertex into the quadratic
relation to determine the y-coordinate of the vertex.
• If you graph y = ax 2 + bx + c using graphing technology, then
you can approximate the vertex from the graph or determine it
exactly, depending on the features of the technology.
Aid
Examples 1 to 3.
• Try Chapter Review
Questions 8 to 10.
Study
Aid
• See Lesson 5.5,
Examples 1, 2, and 4.
• Try Chapter Review
Questions 11 to 13.
Study
Aid
• See Lesson 5.5, Example 3,
and Lesson 5.6,
Examples 1 and 2.
• Try Chapter Review
Questions 14 to 17.
In all cases, after you know the vertex, you can use the value of
a from the standard form of the relation to write the relation in vertex
form, y = a(x - h)2 + k.
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PRACTICE Questions
Lesson 5.1
7. Reggie used transformations to graph
1. Write the equations of two different quadratic
relations that match each description.
a) The graph has a narrower opening than
the graph of y = 2x 2.
b) The graph has a wider opening than
the graph of y = - 0.5x 2.
c) The graph opens downward and has a
narrower opening than the graph of y = 5x 2.
2. The point ( p, q) lies on the parabola y = ax 2. If
you did not know the value of a, how could you
use the values of p and q to determine whether
the parabola is wider or narrower than y = x 2?
3. Match each translation with the correct
8. Use the point marked on each parabola, as well
a)
quadratic relation.
a) 3 units left, 4 units down
b) 2 units right, 4 units down
c) 5 units left
d) 3 units right, 2 units up
b)
c) d)
y
8
4
x
i) y = (x - 3)2 + 2 iii) y = (x - 2)2 - 4
ii) y = (x + 3)2 - 4 iv) y = (x + 5)2
Lesson 5.3
the graph shown? Explain
your reasoning.
a) y = - 3(x + 3)2 + 8
b) y = - 3(x - 3)2 + 8
c) y = 3(x - 3)2 - 8
d) y = - 2(x - 3)2 + 8
Lesson 5.4
as the vertex of the parabola, to determine the
equation of the parabola in vertex form.
Lesson 5.2
4. Which equation represents
y = - 2(x - 4)2 + 3. He started by reflecting
the graph of y = x 2 in the x-axis. Then he
translated the graph so that its vertex moved to
(4, 3). Finally, he stretched the graph vertically
by a factor of 2.
a) Why was Reggie’s final graph not correct?
b) What sequence of transformations should
he have used?
c) Use transformations to sketch
y = - 2(x - 4)2 + 3 on grid paper.
-4
4
8
4
-8
10. This table shows residential energy use by
2
x
0
2
4
6
5. The parabola y = x 2 is transformed in two
Canadians from 2002 to 2006, where
1 petajoule equals 1 000 000 000 000 000 joules.
Year
Residential Energy Use
(petajoules)
different ways to produce the parabolas
y = 2(x - 4)2 + 5 and y = 2(x - 5)2 + 4.
How are these transformations the same, and
how are they different?
2002
1286.70
2003
1338.20
2004
1313.00
6. Blake rotated the parabola y = x 2 by 180° around
2005
1296.60
a point. The new vertex is (6, - 8). What is the
equation of the new parabola?
2006
1250.30
304
Chapter Review
12
equation of each quadratic relation in vertex form.
a) vertex at ( - 3, 2), passes through ( - 1, 4)
b) vertex at (1, 5), passes through (3, - 3)
6
-2
-4
9. Use the given information to determine the
y
8
-12 -8
0
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Chapter Review
a) Use technology to create a scatter plot and
a quadratic regression model.
b) Determine the vertex, and write the
equation of the model in vertex form.
c) According to your model, when was energy
use at a maximum during this period?
Lesson 5.5
11. Karla hits a golf ball from an elevated tee to the
green below. This table shows the height of the
ball above the ground as it moves through the air.
bands and booking DJs. By tracking receipts
over a period of time, the owner of the club
determined that her profit from a live band
depended on the ticket price. Her profit, P, can
be modelled using P = - 15x 2 + 600x + 50,
where x represents the ticket price in dollars.
a) Sketch the graph of the relation to help the
owner understand this profit model.
b) Determine the maximum profit and the
ticket price she should charge to achieve the
maximum profit.
Time (s)
Height (m)
0.0
30.00
0.5
41.25
1.0
50.00
1.5
56.25
2.0
60.00
2.5
61.25
3.0
60.00
Lesson 5.6
3.5
56.25
15. Express each quadratic relation in vertex form using
4.0
50.00
a) Create a scatter plot, and draw a curve of
good fit.
b) Estimate the coordinates of the vertex.
c) Determine a quadratic relation in vertex
form to model the data.
d) Use the quadratic regression feature of
graphing technology to create a model for
the data. Compare this model with the
model you created by hand for part c). How
accurate is the model you created by hand?
12. A farming community collected data on the
effect of different amounts of fertilizer, x, in
100 kg/ha, on the yield of carrots, y, in tonnes.
The resulting quadratic regression model is
y = - 0.5x 2 + 1.4x + 0.1. Determine the
amount of fertilizer needed to produce the
maximum yield.
NEL
13. A local club alternates between booking live
14. For each quadratic relation,
i) write the equation in factored form
ii) determine the coordinates of the vertex
iii) write the equation in vertex form
iv) sketch the graph
a) y = x 2 - 8x + 15
b) y = 2x 2 - 8x - 64
c) y = - 4x 2 - 12x + 7
partial factoring to determine two points that are
the same distance from the axis of symmetry.
a) y = x 2 + 2x + 5
b) y = - x 2 + 6x - 3
c) y = - 3x 2 + 42x - 147
d) y = 2x 2 - 20x + 41
16. Write each quadratic relation in vertex form
using an appropriate strategy.
a) y = x 2 - 6x - 8
b) y = - 2(x + 3)(x - 7)
c) y = x(3x + 12) + 2
d) y = - 2x 2 + 12x - 11
17. The height, h, of a football in metres t seconds
since it was kicked can be modelled by
h = - 4.9t 2 + 22.54t + 1.1.
a) What was the height of the football when
the punter kicked it?
b) Determine the maximum height of the
football, correct to one decimal place, and the
time when it reached this maximum height.
Chapter 5
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y
10
1. a) The black graph at the left resulted from transforming the green
y x2
graph of y = x 2. Determine the equation of the black graph.
Explain your reasoning.
b) State the transformations that were applied to the graph of y = x 2
to result in the black graph.
6
4
2. Determine the equation of each quadratic relation in vertex form.
2
x
-4
-2
0
-2
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Chapter Self-Test
8
-6
9:38 AM
2
4
6
-4
8
a) vertex at (7, 5), opens downward, vertical stretch of 4
b) zeros at 1 and 5, minimum value of - 12, passes through (6, 15)
3. Sketch each quadratic relation by applying the correct sequence of
-6
transformations to the graph of y = x 2.
a) y = - 2(x - 3)2 + 8
b) y = 0.5(x + 2)2 - 5
-8
4. The parabola y = x 2 is compressed vertically and translated down and
-10
right. The point (4, - 10) is on the new graph. What is a possible
equation for the new graph?
5. Accountants for the HiTech Shoe Company have determined that the
Process
Checklist
✔ Question 2: Did you
connect the information
about each parabola to the
appropriate form of the
relation?
✔ Question 4: Did you apply
reasoning skills as you
developed a possible
equation for the graph?
✔ Questions 5 and 6: Did you
reflect on your thinking to
assess the appropriateness
of your strategies as you
solved the problems?
✔ Question 6: Did you relate
the numeric, algebraic,
graphical, and verbal
representations of the
situation?
306
Chapter Self-Test
quadratic relation P = - 2x 2 + 24x - 54 models the company’s profit
for the next quarter. In this relation, P represents the profit (in $100 000s)
and x represents the number of pairs of shoes sold (in 100 000s).
a) Express the equation in factored form.
b) What are the zeros of the relation? What do they represent
in this context?
c) Determine the number of pairs of shoes that the company must sell
to maximize its profit. How much would the maximum profit be?
6. A toy rocket that is sitting on a tower is launched vertically upward.
The table shows the height, h, of the rocket in centimetres at t seconds
after its launch.
t (s)
0
1
2
3
4
5
6
7
h (cm)
88
107
116
115
104
83
52
11
a) Using a graphing calculator, create a scatter plot to display the
data.
b) Estimate the vertex of your model. Then write the equation of the
model in vertex form and standard form.
c) Use the regression feature on the graphing calculator to create a
quadratic model for the data. Compare this model with the model
you created for part b).
d) What is the maximum height of the rocket? When does the rocket
reach this maximum height?
e) When will the rocket hit the ground?
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Chapter Task
Human Immunodeficiency Virus (HIV)
Every year, many people become infected with HIV. Over 90% of HIV
infections in children in the United States are due to mother-to-child
transmission at birth. The data in the table show the number
of mother-to-child HIV infections diagnosed in the United States
in various years from 1985 to 2005.
Year
1985 1987 1990 1993 1996 1998 2001 2003 2005
Number of Cases
210
?
500
780
770
460
300
317 188
142
What can we learn about the fight against HIV infections
from the data?
Health Connection
A.
Create a scatter plot to display the data. Why does a quadratic model
make sense?
B.
Determine an equation for this relation. Which form (standard, vertex,
or factored) do you think is the best for these data? Explain.
Research on HIV focuses on
prevention, and on treatment
and care of people infected.
A mask and gloves are needed
for protection.
C.
Identify the transformations that you would apply to the graph of
y = x 2 to obtain the model you created for part B.
D.
E.
F.
NEL
The decline in the number of mother-to-child HIV infections is due
to the introduction of preventive drug therapies. Based on your model,
when do you think an effective drug therapy was first introduced?
Explain your reasoning.
Based on your model, will the number of HIV cases ever be reduced to
zero? If so, when might this occur? Do you think your prediction is
accurate? Explain your reasoning.
Suggest some reasons why a mathematical description of the data
could be useful to researchers or government agencies.
Task
Checklist
✔ Did you draw your graph
accurately and label it
correctly?
✔ Did you choose an
appropriate graphing
window to display your
scatter plot?
✔ Did you show all the
appropriate calculations?
✔ Did you explain your
reasoning clearly?
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Year
Ontario’s CO2 Emissions
(kilotonnes/year)
1995
175 000
1996
182 000
1997
186 000
1998
187 000
1999
191 000
2000
201 000
2001
193 000
2002
199 000
2003
203 000
2004
199 000
2005
201 000
CO2 emissions are measured in
kilotonnes (kt); 1 kt 1000 tonnes (t)
and 1 t 1000 kg.
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Chapter
6
Quadratic
Equations
GOALS
You will be able to
• Solve quadratic equations graphically, by
factoring, and by using the quadratic
formula
• Write a quadratic relation in vertex form
by completing the square
• Solve and model problems involving
CO2 Emissions (kilotonnes/year)
quadratic relations in standard, factored,
and vertex forms
210 000
Optimistic Model of CO2 Emissions in Ontario
y
200 000
? Recent attention to the environment
190 000
has raised awareness about the
effects of carbon dioxide in the
atmosphere. Many countries are
developing strategies to reduce their
CO2 emissions.
180 000
170 000
160 000
150 000
x
0
1995
NEL
2000
2005
Year
2010
2015
How can you use a quadratic
model to predict when Ontario’s
CO2 emissions might drop below
1995 levels?
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1. State the vertex, equation of the axis of symmetry, and zeros of the
parabola at the left.
4
2. Match each form with the correct equation.
2
x
-2
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WORDS YOU NEED to Know
6
-6 -4
3:49 PM
Getting Started
y
8
5/7/09
0
2
a) standard form
b) factored form
c) vertex form
i) y = - 2(x + 3)(x - 1)
ii) y = - 2(x + 1)2 + 8
iii) y = - 2x 2 - 4x + 6
SKILLS AND CONCEPTS You Need
Graphing Quadratic Relations
Study
Aid
• For more help and practice,
see Lessons 5.6, 3.3,
and 5.3.
Different strategies can be used to graph a quadratic relation. The strategy
you use might depend on the form of the relation.
EXAMPLE
Describe a strategy you could use to graph each quadratic relation.
a) y = x 2 + 4x - 1
b) y = - 2(x + 3)(x - 5)
c) y = 2(x - 3)2 - 4
Solution
a) The equation is in standard form.
• Partially factor the equation to locate two ordered pairs
with the same y-coordinate.
• Determine the x-coordinate of the vertex by calculating the mean
of the x-coordinates of the points you determined above.
• Substitute the x-coordinate of the vertex into the equation
to determine the y-coordinate of the vertex.
• Substitute two other values of x into the equation to determine
two more points on the parabola.
• Use symmetry to determine the points on the parabola that are directly
across from the two additional points you determined.
• Plot the vertex and points, then sketch the parabola.
b) The equation is in factored form.
• Locate the zeros by setting each factor to zero and solving
each equation.
• Determine the x-coordinate of the vertex by calculating the mean
of the x-coordinates of the zeros that you determined above.
• Substitute the x-coordinate of the vertex into the equation
to determine the y-coordinate of the vertex.
• Plot the vertex and zeros, then sketch the parabola.
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Getting Started
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Getting Started
c) The equation is in vertex form.
• Locate the vertex and the axis of symmetry.
• Determine the y-intercept by letting x equal 0.
• Use symmetry to determine the point on the parabola that is directly
across from the y-intercept.
• Plot the vertex and points, then sketch the parabola.
3. Graph each quadratic relation.
a) y = (x + 4)2 - 3
b) y = - 3(x - 3)2 - 1
c) y = (x + 5)(x - 7)
1
d) y = (x - 4)(x - 7)
2
e) y = 2x 2 + x - 1
f ) y = - 3x 2 - 5x
Factoring Quadratic Expressions
You can use a variety of strategies to factor a quadratic expression.
Study
Aid
• For more help and practice,
EXAMPLE
see Lessons 4.2 to 4.6.
Factor. Use an area diagram for part a). Use decomposition for part b).
a) x 2 - 7x - 18
b) 4x 2 + 8x - 5
Solution
a) x 2 - 7x - 18
x
9
x
x2
9x
2
2x
18
This is a trinomial where a = 1 and there are no
common factors. Look for two binomials that each
start with x. To determine the factors, find two
numbers whose product is 18 and whose sum
is 7. The numbers are 9 and 2.
x 2 - 7x - 18 = (x - 9)(x + 2)
b) 4x 2 + 8x - 5
= 4x 2 - 2x + 10x - 5
This is a trinomial where a Z 1 and there are no
common factors. Look for two numbers whose
sum is 8 and whose product is (4)(5) 20.
The numbers are 2 and 10. Use these
to decompose the middle term.
= 4x 2 - 2x + 10x - 5
= 2x(2x - 1) + 5(2x - 1)
Group the terms in pairs, and divide out
the common factors.
= (2x - 1)(2x + 5)
Divide out the common binomial as a common factor.
4. Factor each expression, if possible.
a) x 2 + 8x + 12
b) x 2 - 5x + 6
NEL
c) x 2 + 7x - 30
d) 9x 2 - 30x + 25
e) - 6x 2 - 7x + 24
f ) 2x 2 - x - 5
Chapter 6
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PRACTICE
Aid
Study
5/7/09
• For help, see the Review of
5. Solve each equation.
Essential Skills and
Knowledge Appendix.
a) 4x + 8 = 0
b) 5x - 3 = 0
Question
Appendix
5
A-9
1
x
x
x
x2
x2
x
1
1
1
x
x
x
x
x
x
1
1
1
c) -2x + 12 = 0
d) 12x + 7 = 0
6. Expand and simplify.
a) (3x - 5)(x - 4) c) (2x + 3)(4x - 5) e) (3a + 7)(3a + 7)
b) (n + 1)(n - 1) d) (7 - 3p)(2p + 5) f ) (6x - 5)2
7. The algebra tiles at the left show 2x 2 - 7x + 3 and its factors.
Determine the factors for each expression. Use algebra tiles or area
diagrams, if you wish.
a) x 2 + 4x + 3
c) 3x 2 - 5x - 2
e) 2x 2 + 12x
2
2
b) x - 8x + 16
d) 4x - 9
f ) 9x 2 - 6x + 1
8. For each quadratic relation, determine the zeros, the y-intercept,
the equation of the axis of symmetry, the vertex, and the equation
in standard form.
y
a)
b)
y
28
6
24
4
20
2
x
16
-8 -6 -4
12
8
x
-2
0
-4
-2
2
4
-4
4
-4
-2
0
2
4
6
8
-6
-8
-10
9. For each quadratic relation, determine the y-intercept, the equation
of the axis of symmetry, and the vertex.
a) y = (x - 4)(x + 6)
b) y = - 4(x - 3)2 - 5
10. Do you agree or disagree with each statement? Provide examples
to support your answers.
a) Every quadratic expression can be written as the product
of two linear factors.
b) Every quadratic relation has a maximum value or a minimum value.
c) The graph of a quadratic relation always has two x-intercepts.
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Getting Started
APPLYING What You Know
The Jewel Box
YOU WILL NEED
• grid paper
• ruler
This building is called the Jewel Box. It is a large greenhouse in St. Louis,
Missouri. Its design is based on a parabola that passes through the corners
of the roof line.
?
What quadratic relations can be used to model this parabola?
A.
Trace the parabola from the photo at the right onto grid paper.
Decide where to draw the x- and y-axes.
B.
Create an algebraic model for the parabola in vertex form.
C.
Create an algebraic model for the parabola in factored form.
D.
How are the two models you created for parts B and C the same?
How are they different?
E.
Write both of your models in standard form. Do all three models
represent the same parabola? Explain.
F.
Which form of the quadratic relation do you prefer to model the shape
of the roof line of the Jewel Box? Justify your answer.
NEL
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Solving Quadratic Equations
YOU WILL NEED
GOAL
• grid paper
• ruler
• graphing calculator
Use graphical and algebraic strategies to solve quadratic equations.
INVESTIGATE the Math
Andy and Susie run a custom T-shirt business. From past experience, they
know that they can model their expected profit, in dollars, with the relation
P = - x 2 + 120x - 2000, where x is the number of T-shirts they sell.
Andy wants to sell enough T-shirts to earn $1200. Susie wants to sell just
enough T-shirts to break even because she wants to close the business.
?
How can Andy and Susie determine the number of T-shirts
they must sell to achieve their goals?
quadratic equation
A.
an equation that contains at
least one term whose highest
degree is 2; for example,
x2 + x - 2 = 0
Why can you use the quadratic equation -x 2 + 120x - 2000 = 0
to determine the number of T-shirts that must be sold to achieve
Susie’s goal?
B.
Factor the left side of the equation in part A. Use the factors
to determine the number of T-shirts that must be sold to achieve
Susie’s goal.
C.
Use your factors for part B to predict what the graph of the profit
relation will look like. Sketch the graph, based on your prediction.
D.
Graph the profit relation using a graphing calculator. Was your
prediction for part C correct?
E.
What quadratic equation can you use to describe Andy’s goal
of making a profit of $1200?
F.
How can you use your graph for part D to determine the roots
of your equation for part E?
G.
How many T-shirts must be sold to achieve Andy’s goal?
root
a solution; a number that can be
substituted for the variable to
make the equation a true
statement; for example, x = 1 is
a root of x 2 + x - 2 = 0, since
12 + 1 - 2 = 0
314
Reflecting
H.
Why did factoring -x 2 + 120x - 2000 help you determine
the break-even points?
I.
Are the roots of the equation -x 2 + 120x - 2000 = 0 also zeros
or x-intercepts of the relation y = - x 2 + 120x - 2000? Explain.
6.1 Solving Quadratic Equations
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6.1
J.
Why would factoring the left side of -x 2 + 120x - 2000 = 1200
not help you determine the number of T-shirts that Andy has to sell?
K.
Explain why it would help you solve the equation in part J if you were
to write it as - x 2 + 120x - 2000 - 1200 = 0.
L.
To solve ax + b = c, you isolate x. Why would you not isolate x 2
to solve ax 2 + bx + c = 0?
APPLY the Math
EXAMPLE 1
Selecting a strategy to solve a quadratic
equation
The user’s manual for Arleen’s model rocket says that the equation
h = - 5t 2 + 40t models the approximate height, in metres,
of the rocket after t seconds. When will Arleen’s rocket reach a height
of 60 m?
Amir’s Solution: Selecting a factoring strategy
- 5t 2 + 40t = 60
- 5t 2 + 40t - 60 = 0
- 5(t 2 - 8t + 12) = 0
- 5(t - 2)(t - 6) = 0
t - 2 = 0 or t - 6 = 0
t = 2
t = 6
The rocket is 60 m above the ground
at 2 s on the way up and 6 s on the
way down.
NEL
I substituted 60 for h because
I wanted to calculate the time
for the height 60 m.
I subtracted 60 from both sides
of the equation to make the
right side equal zero. I did this
so that I could determine the
zeros of the corresponding
relation.
I divided out the common factor
of 5. Then I factored the
trinomial. The trinomial will
equal zero if either factor equals
zero. I set each factor equal to
zero and solved both equations.
This gave me the zeros of the
parabola.
Chapter 6
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Support
For help locating the zeros
of a relation using a TI 83/84
graphing calculator, see
Appendix B-8. If you are using
a TI-nspire, see Appendix B-44.
I verified my solutions by
graphing y = - 5x2 + 40x - 60,
and then locating the zeros.
My solutions were correct.
Alex’s Solution: Selecting a graphing strategy
Using a graphing calculator,
I entered the height equation in
Y1. I substituted the variable y
for h and the variable x for t.
I entered 60 in Y2 to determine
when the rocket will reach 60 m.
I estimated that the rocket will
travel about 100 m and be in the
air for about 10 s, so I used
these window settings.
I used the Intersect operation
to locate the intersection points
of the two graphs.
Tech
Support
For help determining points of
intersection using a TI-83/84
graphing calculator, see
Appendix B-11. If you are using
a TI-nspire, see Appendix B-47.
The rocket is 60 m off the ground
after 2 s and after 6 s.
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6.1 Solving Quadratic Equations
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6.1
EXAMPLE 2
Selecting a factoring strategy to solve a quadratic equation
Determine the roots of 6x 2 - 11x - 10 = 0.
Annette’s Solution
6x 2 - 11x - 10 = 0
Product 60 Sum 11
(1)(60)
1 (60) 59✘
(2)(30)
2 (30) 28✘
(3)(20)
3 (20) 17✘
(4)(15)
4 (15) 11✔
6x 2 - 15x + 4x - 10 = 0
3x(2x - 5) + 2(2x - 5) = 0
(2x - 5)(3x + 2) = 0
2x - 5 = 0 or 3x + 2 = 0
2x = 5
3x = - 2
2
5
x = x =
2
3
Since the trinomial in the equation contains no
common factors and is one where a Z 1, I used
decomposition. I looked for two numbers whose
sum is 11 and whose product is (6)(10) 60.
Since the numbers were 15 and 4, I used these
to decompose the middle term. I factored the first
two terms and then the last two terms. Then,
I divided out the common factor of 2x - 5.
I set each factor equal to zero and solved
each equation.
The roots of 6x 2 - 11x - 10 = 0
1
2
are x = 2 and x = - .
2
3
EXAMPLE 3
Reasoning about how to solve a quadratic equation
Determine all the values of x that satisfy the equation x 2 + 4 = 3x(x - 5).
If necessary, round your answers to two decimal places.
Karl’s Solution
x 2 + 4 = 3x(x - 5)
x 2 + 4 = 3x 2 - 15x
0 = 3x 2 - x 2 - 15x - 4
0 = 2x 2 - 15x - 4
NEL
I decided to write an equivalent equation in the form
ax 2 + bx + c = 0, which I could solve by graphing
or factoring. I expanded the expression on the right
side of the equation.
I used inverse operations to make the left side of the
equation equal to zero. I couldn’t factor the right side
of the equation, so I decided to use a graph.
Chapter 6
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I graphed y = 2x 2 - 15x - 4 using these window
settings. From the graph, I could see that one
x-intercept was between 1 and 0 and the other
x-intercept was between 7 and 8.
Using the Zero operation of the calculator, I estimated
that the x-intercepts were about 0.258 and 7.758.
Tech
Support
For help determining the zeros
of a relation using a TI-83/84
graphing calculator, see
Appendix B-8. If you are using
a TI-nspire, see Appendix B-44.
The solutions are x - 0.26 and x 7.76.
EXAMPLE 4
I rounded the solutions to two decimal places.
These are reasonable estimates, since the solutions
are not exact.
Reflecting on the reasonableness of a solution
A ball is thrown from the top of a seaside cliff. Its height, h, in metres,
above the sea after t seconds can be modelled by h = - 5t 2 + 21t + 120.
How long will the ball take to fall 20 m below its initial height?
Jacqueline’s Solution
h = - 5t 2 + 21t + 120
h = - 5(0)2 + 21(0) + 120
h = 120
The cliff is 120 m high, so the ball starts
120 m above the sea.
120 - 20 = 100
Let h = 100.
100 = - 5t 2 + 21t + 120
0 = - 5t 2 + 21t + 120 - 100
0 = - 5t 2 + 21t + 20
318
6.1 Solving Quadratic Equations
I let t 0 to determine the initial height of the ball.
The initial height of the ball was 120 m. When
the ball had fallen 20 m below its initial height,
it was 100 m above the sea.
I substituted 100 for h in the relation. I wrote the
equation in the form 0 = ax 2 + bx + c so that
I could solve it by graphing or factoring.
I subtracted 100 from both sides of the equation
to make the left side equal to 0.
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6.1
0 = 5t 2 - 21t - 20
0 = 5t 2 - 25t + 4t - 20
0 = 5t(t - 5) + 4(t - 5)
0 = (5t + 4)(t - 5)
5t + 4 = 0 or
5t = - 4
4
t = 5
t - 5 = 0
t = 5
The ball will take 5 s to fall 20 m below
its initial height.
I multiplied all the terms, on both sides of the
equation, by 1 because I wanted 5t 2 to be
positive. I factored the right side of the equation
using decomposition.
I set each factor equal to zero and solved for t.
Since the ball was thrown at t = 0, I knew that the
4
solution t = - didn’t make sense. I used the
5
solution t = 5 since this did make sense.
In Summary
Key Ideas
• A quadratic equation is any equation that contains a polynomial in
one variable whose degree is 2; for example, x 2 + 6x + 9 = 0.
• All quadratic equations can be expressed in the form ax 2 + bx + c = 0
using algebraic strategies. In this form, the equation can be solved by
• factoring the quadratic expression, setting each factor equal to zero,
and solving the resulting equations
or
• graphing the corresponding relation y = ax 2 + bx + c and
determining the zeros, or x-intercepts
Need to Know
• Roots and solutions have the same meaning. These are all values
that satisfy an equation.
• Some quadratic equations can be solved by factoring. Other quadratic
equations must be solved by using a graph.
• If you use factoring to solve a quadratic equation, write the equation
in the form ax 2 + bx + c = 0 before you try to factor.
• To solve ax 2 + bx + c = d using a graph, graph y = ax 2 + bx + c
and y = d on the same axes. The solutions to the equation are the
x-coordinates of the points where the parabola and the horizontal
line intersect.
CHECK Your Understanding
1. The solutions to each equation are the x-intercepts of the
corresponding quadratic relation. State the quadratic relation.
a) x 2 - 4x + 4 = 0
b) 2x 2 - 9x = 5
NEL
Chapter 6
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2. Use the graph of each quadratic relation to determine the roots
to each quadratic equation, where y = 0.
y
a)
b)
8
4
x
-6 -4 -2
0
-4
-8
2
4
6
-6 -4 -2
y
8 y 2x2 5x 3
4
x
0
-4
2
4
6
-8
y x2 9
3. Solve each equation.
a) x(x + 4) = 0
b) (x + 10)(x + 8) = 0
c) (x - 5)2 = 0
d) (3x + 8)(x - 4) = 0
e) x 2 + 5x + 6 = 0
f ) x 2 - 2x = 8
PRACTISING
4. Determine whether the given value is a root of the equation.
a) x = 2; x 2 + x - 6 = 0
3
; 8x 2 + 10x - 3 = 0
2
e) x = - 5; x 2 - 4x - 5 = 0
d) x =
b) x = 4; x 2 + 7x - 8 = 0
1
c) x = - ; 2x 2 + 11x + 5 = 0 f ) x = 2; 3x 2 - 2x - 8 = 0
2
5. Solve each equation by factoring. Use an equivalent equation, if necessary.
a) x 2 + 2x - 15 = 0
b) x 2 + 5x - 24 = 0
c) x 2 + 4x + 4 = 0
d) x 2 - 5x = 0
e) x 2 - 6x = 16
f ) x 2 + 12 = 7x
6. Solve by factoring. Verify your solutions.
a) 3x 2 - 5x - 2 = 0
b) 2x 2 + 3x - 2 = 0
c) 3x 2 - 4x - 15 = 0
d) 6x 2 - x - 2 = 0
e) 4x 2 - 4x = 3
f ) 9x 2 + 1 = 6x
7. Simplify and then solve each equation.
K
a) x(x + 1) = 12
b) 2x(x + 4) = x + 4
c) 3x(x + 2) = 2x 2 - (4 - x)
d) 3x(x + 6) + 50 = 2x 2 + 3(x - 2)
e) (x + 2)2 + x = 2(3x + 5)
f ) (2x + 1)2 = x + 2
8. Determine the roots of each equation.
a) x 2 + 4x - 32 = 0
b) x 2 + 11x + 30 = 0
c) 5x 2 - 28x - 12 = 0
d) x 2 + 5x = 14
e) 4x 2 + 25 = 20x
f ) 3x 2 + 16x - 7 = 5
9. Solve each equation. Round your answers to two decimal places.
a) x 2 + 5x - 2 = 0
b) 4x 2 - 8x + 3 = 0
c) x 2 + 1 = 4 - 2x 2
320
6.1 Solving Quadratic Equations
d) x(x + 5) = 2x + 7
e) 3x 2 + 5x - 3 = x 2 + 4x + 1
f ) (x + 3)2 - 2x = 15
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6.1
10. Conor has a summer lawn-mowing business. Based on experience,
Conor knows that P = - 5x 2 + 200x - 1500 models his profit, P,
in dollars, where x is the amount, in dollars, charged per lawn.
a) How much does he need to charge if he wants to break even?
b) How much does he need to charge if he wants to have a profit
of $500?
11. Stacey maintains the gardens in the city parks. In the summer, she
plans to build a walkway through the rose garden. The area of the
walkway, A, in square metres, is given by A = 160x + 4x 2, where x is
the width of the walkway in metres. If the area of the walkway must be
900 m2, determine the width.
12. Patrick owns an apartment building. He knows that the money he
earns in a month depends on the rent he charges. This relationship
1
can be modelled by E =
R (1650 - R), where E is Patrick’s
50
monthly earnings, in dollars, and R is the amount of rent, in dollars,
he charges each tenant.
a) How much will he earn if he sets the rent at $900?
b) If Patrick wants to earn at least $13 000, between what two values
should he set the rent?
Environment Connection
By photosynthesis, green plants
remove carbon dioxide from
the air and produce oxygen.
13. Determine the points of intersection of the line y = - 2x + 7 and
T
the parabola y = 2x 2 + 3x - 5.
14. While hiking along the top of a cliff, Harlan knocked a pebble over
A
the edge. The height, h, in metres, of the pebble above the ground
after t seconds is modelled by h = - 5t 2 - 4t + 120.
a) How long will the pebble take to hit the ground?
b) For how long is the height of the pebble greater than 95 m?
15. Is it possible to solve a quadratic equation that is not factorable over
C
the set of integers? Explain.
16. a) Describe when and why you would rewrite a quadratic equation
to solve it. In your answer, include x 2 - 2x = 15, rewritten as
x 2 - 2x - 15 = 0.
b) Explain how the relation y = x 2 - 2x - 15 can be used to solve
x 2 - 2x - 15 = 0.
Extending
17. Solve the equations x 4 - 9x 2 + 20 = 0 and x 3 - 9x 2 + 20x = 0
by first solving the equation x 2 - 9x + 20 = 0.
18. Will all quadratic equations always have two solutions? Explain how
you know and support your claim with examples.
NEL
Chapter 6
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Exploring the Creation
of Perfect Squares
YOU WILL NEED
GOAL
• algebra tiles
Recognize the relationship between the coefficients and constants
of perfect-square trinomials.
EXPLORE the Math
Quadratic expressions like x 2 + 8x + 16 and 4x 2 + 8x + 4
are perfect-square trinomials. Quadratic expressions like 4x 2 + 8x + 3
are not.
322
x 2 + 8x + 16
4x 2 + 8x + 4
x2
x x x x
x2
x2
x x
x2
x2
x x x
x
x
x
x
1
1
1
1
x2
x2
x x
x2
x2
x x x
x
x
x
x
1
1
x
x
1
1
1
1
1
1
1
1
1
1
1
1
1
4x 2 + 8x + 3
1
1
1
1
?
How can you decide what value for c makes expressions of the
form ax2 + abx + c, a Z 0, perfect-square trinomials?
A.
Factor x 2 + 8x + 16 and 4x 2 + 8x + 4 completely. Explain why
these expressions are called perfect-square trinomials.
B.
Using algebra tiles, create an arrangement that helps you determine the
constant term c that must be added to create perfect-square trinomials.
Verify by factoring each new trinomial you created.
i) x 2 + 2x + c iii) x 2 + 6x + c
v) x 2 + 10x + c
2
2
ii) x + 4x + c iv) x + 8x + c
vi) x 2 + 12x + c
C.
For each trinomial you created in part B, compare the coefficient of x and
the constant term you added. Explain how these numbers are related.
D.
How are the expressions below different from those in part B?
i) x 2 - 4x + c
iii) x 2 - 6x + c
v) x 2 - 2x + c
ii) x 2 - 8x + c
iv) x 2 - 12x + c
vi) x 2 - 10x + c
E.
Using algebra tiles or an area diagram, determine the constant term c
that must be added to each of the expressions in part D to create
perfect-square trinomials. Verify by factoring each new trinomial.
6.2 Exploring the Creation of Perfect Squares
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6.2
F.
For each trinomial you created for part E, compare the coefficient of x
and the constant term you added. Does the relationship you
discovered in part C still apply?
G.
Each expression below contains a common factor. Factor the
expression and then determine the constant term c that must be
added to each expression to make it a multiple of a perfect-square
trinomial. Verify by factoring each new trinomial.
i) 2x 2 + 4x + c
iii) 3x 2 - 6x + c
v) 5x 2 + 25x + c
2
2
ii) 3x - 12x + c
iv) -x + 4x + c vi) 6x 2 + 54x + c
Reflecting
H.
How can you predict the value of c that will make x 2 + bx + c
a perfect-square trinomial?
I.
How can you predict the value of c that will make ax 2 + abx + c
a perfect-square trinomial?
In Summary
Key Idea
b 2
b 2
b = x2 + bx + a b . So,
2
2
in all perfect-square trinomials, the constant term is half the coefficient
of the x term squared.
• If (x + b)2 = x2 + 2bx + b2, then ax +
Need to Know
• To create a perfect square that includes x 2 + bx and no other terms
b 2
with a variable, add a b .
2
• To create a perfect square that includes ax 2 + abx and no other terms
with a variable, factor out a and then create a perfect square that
b 2
includes x 2 + bx. This results in adding aa b .
2
FURTHER Your Understanding
1. Determine the value of c that will create a perfect-square trinomial.
Verify by factoring the trinomial you created.
a) x 2 + 8x + c
c) x 2 + 40x + c
2
b) x - 14x + c
d) x 2 + 20x + c
e) x 2 - 5x + c
f) x 2 + x + c
2. Each expression is a multiple of a perfect-square trinomial. Determine
the value of c.
a) 3x 2 + 30x + c
b) 2x 2 - 12x + c
NEL
c) - 4x 2 - 8x + c
d) 6x 2 - 60x + c
e) 5x 2 - 10x + c
f ) 7x 2 + 42x + c
Chapter 6
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Curious Math
Does 65 Equal 64?
Let a 1 and b 1.
So a b.
a * a = a * b
1. Copy the steps. Explain how each
a 2 = ab
step is obtained from the step
2
a - b 2 = ab - b 2
above it.
(a + b)(a - b) = b(a - b)
2. Can you find any problems with any
a + b = b
of the steps?
2 = 1
2 + 63 = 1 + 63
65 = 64
The steps at the right seem to prove that
65 equals 64.
The two diagrams below also seem to prove that 65 equals 64.
3. How do the colours make the rectangle and the square appear to have
the same area?
4. Determine the area of each figure.
5. Use your answers for steps 3 and 4 to explain why these two figures
appear to prove that 65 equals 64.
6. These two proofs are called fallacious proofs because they contain an
error. How would mathematics and our daily lives be affected if either
of these proofs were true?
7. Some fallacious proofs are very complex. Try to create or research
another fallacious proof that you can explain to a classmate.
324
Curious Math
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Completing the Square
GOAL
YOU WILL NEED
Write the equation of a parabola in vertex form by completing
the square.
• algebra tiles (optional)
• grid paper
• ruler
h
LEARN ABOUT the Math
8
The automated hose on an aerial ladder sprays water on a forest fire.
The height of the water, h, in metres, can be modelled by the relation
h = - 2.25x 2 + 4.5x + 6.75, where x is the horizontal distance, in
metres, of the water from the nozzle of the hose.
?
6
4
How high did the water spray from the hose?
EXAMPLE 1
Selecting a strategy to solve a problem
2
- 2.25x 2 + 4.5x + 6.75 in vertex form by
Write the height relation h =
completing the square to determine the maximum height.
Joan’s Solution: Selecting algebra tiles to complete the square
y = a(x - h)2 + k
I knew that the vertex and
maximum value can be
determined from an equation in
vertex form. I also knew that the
value of a is the same in both
standard form and vertex form.
h = - 2.25x 2 + 4.5x + 6.75
h = - 2.25(x 2 - 2x - 3)
I factored out 2.25 to get a
trinomial with integer coefficients
that I might be able to factor.
y = ax 2 + bx + c
1
x2
x
1 1
NEL
x
1
1
Because I wanted the equation
in vertex form, I tried to make
x 2 - 2x - 3 into a perfect
square using tiles. I needed 1
positive unit tile in the corner
to create a perfect square.
I had 3 negative unit tiles,
so I added 1 zero pair.
x
-2
0
2
Career Connection
As well as fighting fires,
firefighters are trained to
respond to medical and
accident emergencies.
completing the square
a process used to rewrite
a quadratic relation that
is in standard form,
y = ax 2 + bx + c, in
its equivalent vertex form,
y = a(x - h)2 + k
Chapter 6
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1
x2
x
x
1
1
1
1
h = - 2.25 C(x - 1)2 - 4 D
I arranged the tiles to make the
perfect square x 2 - 2x + 1.
I had four negative unit tiles
left over. This showed that
x 2 - 2x - 3 = (x - 1)2 - 4.
h = - 2.25 C (x - 1)2 - 4 D
h = - 2.25(x - 1)2 - (- 2.25)(4)
h = - 2.25(x - 1)2 + 9
To write the equation in vertex
form, I multiplied by 2.25 using
the distributive property.
The water sprayed to a maximum
height of 9 m above the ground.
The vertex of the parabola is (1, 9),
and a 6 0. So, the y-coordinate
of the vertex gives the maximum
height of the water.
Arianna’s Solution: Selecting an algebraic strategy
to complete the square
h = - 2.25x 2 + 4.5x + 6.75
h = - 2.25(x 2 - 2x) + 6.75
Since a 6 0, the parabola opens
downward and the maximum
height is the y-coordinate of
the vertex. I had to write the
equation in vertex form.
To do so, I needed to create
a perfect-square trinomial that
used the variable x. I started by
factoring out the coefficient of
x 2 from the first two terms, since
a perfect square can be created
using the x 2 and x terms.
2
= - 1 and (-1)2 = 1,
2
so x 2 - 2x + 1 = (x - 1)2
To create a perfect square,
the constant term had to be the
square of half the coefficient
of the x term.
- 2.25(x 2 - 2x + 1 - 1) + 6.75
I knew that if I added 1 in
the brackets, I would have to
subtract 1 so that I did not
change the equation.
-
h =
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6.3 Completing the Square
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h = - 2.25 C(x 2 - 2x + 1) - 1 D + 6.75
6.3
h = - 2.25 C(x - 1)2 - 1 D + 6.75
I factored the perfect
square.
h = - 2.25(x - 1)2 - ( -2.25)(1) + 6.75
h = - 2.25(x - 1)2 + 9
I multiplied by 2.25 and
collected like terms.
The vertex is (1, 9), so the water sprayed
to a maximum height of 9 m above the ground.
Reflecting
A.
Why did both Joan and Arianna factor out –2.25 first?
B.
Whose strategy do you prefer? Why?
C.
Explain how both strategies involve completing a square.
APPLY the Math
EXAMPLE 2
Connecting a model to the algebraic process of completing the square
Write y = x 2 + 6x + 2 in vertex form, and then graph the relation.
Anya’s Solution
1
x2
x
x
To write the relation in vertex form, I decided to
complete the square using algebra tiles. Since there
was only one x 2 tile, I had to make only one square.
1
x
x
x
x
y = x 2 + 6x + 2
1
x2
1
x x x
x
x
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
I tried to form a square from the tiles, but I didn’t
have enough unit tiles. I needed 9 positive unit tiles
to complete the square.To keep everything balanced,
I added 9 zero pairs.
y = x 2 + 6x + 9 - 9 + 2
NEL
Chapter 6
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x2
x x x
x
x
x
1
1
1
1
1
1
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1
1
1
1
1
1
1
1
1
1
1
1
Page 328
I completed the square. I had (x 3) for its length,
with 2 positive unit tiles and 9 negative unit tiles left
over. Since I could form 2 zero pairs, I had
7 negative unit tiles left over.
y = (x 2 + 6x + 9) - 9 + 2
y = (x + 3)2 - 9 + 2
y = (x + 3)2 - 7
y = x 2 + 6x + 2 in vertex form is
y = (x + 3)2 - 7.
From the algebra tile model, I was able to write
the relation in vertex form.
The vertex is (3, 7).
Since a 7 0, the parabola opens upward.
The equation of the axis of symmetry is x = - 3.
Using the vertex form of the equation, I determined
the vertex, the direction of opening, and the
equation of the axis of symmetry.
y = (0 + 3)2 - 7
y = 9 - 7
y = 2
I let x 0 to determine the y-intercept.
The y-intercept is 2.
y
8
6
4
2
x
-8 -6 -4
-2
0
-2
-4
-6
x 3
328
2
4
6
I plotted the vertex and the y-intercept. I used
symmetry to determine that (6, 2) is also a point
on the parabola.
y (x 3)2 7
-8
6.3 Completing the Square
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6.3
EXAMPLE 3
Solving a problem with an area diagram to complete the square
Cassidy’s diving platform is 6 ft above the water. One of her dives can
be modelled by the equation d = x 2 - 7x + 6, where d is her position
relative to the surface of the water and x is her horizontal distance from
the platform. Both distances are measured in feet. How deep did Cassidy
go before coming back up to the surface?
Sefu’s Solution: Using an area diagram
Since the relation is quadratic and a 7 0, Cassidy’s
deepest point will be at the vertex of a parabola that
opens upward. I decided to complete the square.
d = x 2 - 7x + 6
?
x
7x 6
x2
x
To complete the square, I drew a square area diagram.
I knew that the length and width would have to be
the same and that they both would be x plus
a positive or negative constant.
?
x
3.5
x
x2
3.5x
3.5
3.5x
x
6
Since the middle term 7x had to be split equally
between the two sides of the square, I found the
constant by dividing the coefficient of the middle
term by 2.
-7
- 3.5
2
3.5
x
x2
3.5x
3.5
3.5x
12.25
The constant term in the original equation is 6, but
when I multiplied the 3.5s together to create the
perfect square, I got 12.25, so I had to subtract 6.25.
6.25
6
d = x 2 - 7x + 6
d = (x - 3.5)2 - 6.25
I used the dimensions of my square along with
the extra 6.25 to write the equivalent relation
in vertex form.
Cassidy dove to a depth of 6.25 ft
before turning back toward the surface.
The vertex is (3.5, - 6.25) and since a 7 0, the
y-coordinate of the vertex is her lowest point.
NEL
Chapter 6
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Solving a problem by determining the maximum value
Christopher threw a football. Its height, h, in metres, after t seconds can
be modelled by h = - 4.9t 2 + 11.76t + 1.4. What was the maximum
height of the football, and when did it reach this height?
Macy’s Solution
Since the relation is quadratic, the maximum value
occurs at the vertex. To determine this value, I had to
write the equation in vertex form. I started by
factoring out 4.9 from the first two terms.
h = - 4.9t 2 + 11.76t + 1.4
h = - 4.9(t 2 - 2.4t) + 1.4
- 2.4
= - 1.2 and (- 1.2)2 = 1.44
2
To determine the constant I had to add to t 2 - 2.4t
to create a perfect square, I divided the coefficient of
t by 2. Then I squared my result.
h = - 4.9(t 2 - 2.4t + 1.44 - 1.44) + 1.4
h = - 4.9[(t 2 - 2.4t + 1.44) - 1.44] + 1.4
h = - 4.9[(t - 1.2)2 - 1.44] + 1.4
I completed the square by adding and subtracting
1.44, so the value of the expression value did not
change. I grouped the three terms that formed the
perfect square. Then I factored.
h = - 4.9(t - 1.2)2 + 7.056 + 1.4
h = - 4.9(t - 1.2)2 + 8.456
The vertex is (1.2, 8.456).
I multiplied by 4.9 using the distributive property.
Then I added the constant terms.
The football reached a maximum height
of 8.456 m after 1.2 s.
Since a 6 0, the y-coordinate of the vertex is the
maximum value. The x-coordinate is the time when
the maximum value occurred.
In Summary
Key Idea
• A quadratic relation in standard form, y = ax 2 + bx + c, can be
rewritten in its equivalent vertex form, y = a(x - h)2 + k, by creating
a perfect square within the expression and then factoring it. This
technique is called completing the square.
Need to Know
• When completing the square, factor out the coefficient of x 2 from the
terms that contain variables. Then divide the coefficient of the x term by 2
and square the result. This tells you what must be added and subtracted
to create an equivalent expression that contains a perfect square.
• Completing the square can be used to determine the vertex of a
quadratic relation in standard form.
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6.3
CHECK Your Understanding
1. Copy and replace each symbol to complete the square.
a) y = x 2 + 12x + 5
y = x 2 + 12x + . - . + 5
y = (x 2 + 12x + . ) - . + 5
y = (x + )2 - b) y = 4x 2 + 24x - 15
y = 4(x 2 + . x) - 15
y = 4(x 2 + 6x + - ) - 15
y = 4[(x 2 + 6x + ) - ] - 15
y = 4(x + )2 - . - 15
y = 4(x + )2 - 2. Write each relation in vertex form by completing the square.
a) y = x 2 + 8x
b) y = x 2 - 12x - 3
c) y = x 2 + 8x + 6
3. Complete the square to state the coordinates of the vertex of each relation.
a) y = 2x 2 + 8x
b) y = - 5x 2 - 20x + 6
c) y = 4x 2 - 10x + 1
PRACTISING
4. Consider the relation y = - 2x 2 + 12x - 11.
a) Complete the square to write the relation in vertex form.
b) Graph the relation.
5. Determine the maximum or minimum value of each relation
by completing the square.
a) y = x 2 + 14x
b) y = 8x 2 - 96x + 15
c) y = - 12x 2 + 96x + 6
d) y = - 10x 2 + 20x - 5
e) y = - 4.9x 2 - 19.6x + 0.5
f ) y = 2.8x 2 - 33.6x + 3.1
6. Complete the square to express each relation in vertex form.
K
Then graph the relation.
a) y = x 2 + 10x + 20
b) y = - x 2 + 6x - 1
c) y = 2x 2 + 4x - 2
d) y = - 0.5x 2 - 3x + 4
7. Complete the square to express each relation in vertex form. Then
describe the transformations that must be applied to the graph
of y = x 2 to graph the relation.
a) y = x 2 - 8x + 4
d) y = - 3x 2 + 12x - 6
b) y = x 2 + 12x + 36
e) y = 0.5x 2 - 4x - 8
2
c) y = 4x + 16x + 36
f ) y = 2x 2 - x + 3
8. Joan kicked a soccer ball. The height of the ball, h, in metres, can be
A
NEL
modelled by h = - 1.2x 2 + 6x, where x is the horizontal distance,
in metres, from where she kicked the ball.
a) What was the initial height of the ball when she kicked it?
How do you know?
b) Complete the square to write the relation in vertex form.
c) State the vertex of the relation.
d) What does each coordinate of the vertex represent in this situation?
e) How far did Joan kick the ball?
Health Connection
An active lifestyle contributes
to good physical and mental
health.
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9. Carly has just opened her own nail salon. Based on experience, she
knows that her daily profit, P, in dollars, can be modelled by the relation
P = - 15x 2 + 240x - 640, where x is the number of clients per day.
How many clients should she book each day to maximize her profit?
10. The cost, C, in dollars, to hire landscapers to weed and seed a local
park can be modelled by C = 6x 2 - 60x + 900, where x is the
number of landscapers hired to do the work. How many landscapers
should be hired to minimize the cost?
y 2x2 16x 7
11. Neilles determined the vertex of a relation by completing the square,
as shown at the left. When he checked his answer at the back of his
textbook, it did not match the answer given. Identify each mistake that
he made, explain why it is a mistake, and provide the correct solution.
y 2(x2 8x) 7
y 2(x2 8x 64 64) 7
y 2(x 8)2 64 7
12. Bob wants to cut a wire that is 60 cm long into two pieces. Then he
y 2(x 8)2 73
T
Therefore, the vertex is (73, 8).
wants to make each piece into a square. Determine how the wire should
be cut so that the total area of the two squares is as small as possible.
13. Kayli wants to build a parabolic bridge over a stream in her backyard as
shown at the left. The bridge must span a width of 200 cm. It must be at
least 51 cm high where it is 30 cm from the bank on each side. How
high will her bridge be?
14. a) Determine the vertex of the quadratic relation y = 2x 2 - 4x + 5
51 cm
30 cm
C
200 cm
by completing the square.
b) How does changing the value of the constant term in the relation
in part a) affect the coordinates of the vertex?
15. The main character in a video game, Tammy, must swing on a vine to
cross a river. If she grabs the vine at a point that is too low and swings
within 80 cm of the surface of the river, a crocodile will come out of the
river and catch her. From where she is standing on the riverbank,
Tammy can reach a point on the vine where her height above the river,
h, is modelled by the relation h = 12x 2 - 76.8x + 198, where x is
the horizontal distance of her swing from her starting point. Should
Tammy jump? Justify your answer.
16. Explain how to determine the vertex of y = x 2 - 2x - 35 using three
different strategies. Which strategy do you prefer? Explain your choice.
Extending
17. Celeste has just started her own dog-grooming business. On the first day,
she groomed four dogs for a profit of $26.80. On the second day, she
groomed 15 dogs for a profit of $416.20. She thinks that she will maximize
her profit if she grooms 11 dogs per day. Assuming that her profit can be
modelled by a quadratic relation, calculate her maximum profit.
18. Complete the square to determine the vertex of y = x 2 + bx + c.
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6.3 Completing the Square
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
How are the roots of a quadratic equation and the zeros
of a quadratic relation related?
A:
The roots of the equation ax 2 + bx + c = 0 are the zeros, or
x-intercepts, of the relation y = ax 2 + bx + c.
Q:
What strategies can you use to solve a quadratic equation?
A1:
If the equation is in the form ax 2 + bx + c = 0, you can graph the
relation y = ax 2 + bx + c and locate the zeros on your graph. If the
trinomial is factorable, you can factor it, set each factor equal to zero,
and solve the equations.
Study
Aid
• See Lesson 6.1,
Examples 1 and 2.
• Try Mid-Chapter Review
Questions 1 to 5.
EXAMPLE
Solve x 2 + 2x - 15 = 0.
Solution
By Graphing Technology
or
Graph using a scale - 10 … x … 10
and - 10 … y … 10.
By Factoring
x 2 + 2x - 15 = 0
(x - 3)(x + 5) = 0
x - 3 = 0 or x + 5 = 0
x = 3
x = -5
x = 3 and x = - 5
x = - 5 and x = 3
A2:
If the equation is in the form ax 2 + bx + c = d, you can graph
y = ax 2 + bx + c and y = d and determine the points of intersection.
Alternatively, you can rearrange the equation so that one side is equal to
zero. Then you can graph or factor the resulting equation to solve it.
EXAMPLE
Solve x 2 + 6x + 5 = - 3.
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Solution
y = x 2 + 6x + 5
y = -3
x 2 + 6x + 5 = - 3
x 2 + 6x + 8 = 0
or
x = - 4 and x = - 2
Study
Aid
x 2 + 6x + 5 = - 3
x 2 + 6x + 8 = 0
(x + 4)(x + 2) = 0
or x + 2 = 0
x + 4 = 0
x = -4
x = -2
x = - 4 and x = - 2
x = - 4 and x = - 2
Q:
How can you change a quadratic relation from standard
form to vertex form?
A:
To write a quadratic relation in vertex from, complete the square
as shown below.
• See Lesson 6.3,
Examples 1 to 4.
or
• Try Mid-Chapter Review
Questions 7 to 10.
EXAMPLE
Write the equation in vertex form.
y = 2x 2 - 9x + 2.5
Solution
• When the coefficient of x 2 is a number other
than 1, factor it from the x 2 and x terms. This
will leave a binomial inside the brackets.
• To complete the square for the binomial, add
and subtract the square of half the coefficient
of the x term.
• Group together the three terms that form
the perfect square, and factor it.
y = 2x 2 - 9x + 2.5
y = 2(x 2 - 4.5x) + 2.5
• Use the distributive property to multiply.
Then combine the constants.
y = 2[(x - 2.25)2 - (5.0625)] + 2.5
y = 2(x - 2.25)2 - 2(5.0625) + 2.5
y = 2(x - 2.25)2 - 10.125 + 2.5
y = 2(x - 2.25)2 - 7.625
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Mid-Chapter Review
y = 2(x 2 - 4.5x + 2.252 - 2.252) + 2.5
y = 2(x 2 - 4.5x + 5.0625 - 5.0625) + 2.5
y = 2[(x 2 - 4.5x + 5.0625) - 5.0625] + 2.5
y = 2[(x - 2.25)2 - 5.0625] + 2.5
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Mid-Chapter Review
PRACTICE Questions
Lesson 6.1
Lesson 6.2
1. Solve each quadratic equation.
6. Determine the value of c needed to create
a) x 2 + 4x = 12
b) x 2 + 8x + 9 = 0
c) x 2 - 9x = - 4
d) - 3x 2 - 2x + 3 = 0
e) 2x 2 - 5x + 10 = 15
1
f ) x 2 + 10x - 2 = - 10
2
a perfect-square trinomial.
a) x 2 + 8x + c
b) x 2 - 10x + c
c) x 2 + 5x + c
d) x 2 - 7x + c
e) -4x 2 + 24x + c
f ) 2x 2 - 18x + c
Lesson 6.3
2. Determine the roots.
x 2 + 6x - 16 = 0
a)
b) 2x 2 + x - 3 = 0
c) x 2 + 3x - 10 = 0
d) 6x 2 + 7x - 5 = 0
e) - 3x 2 - 9x + 12 = 0
1
f ) x 2 + 6x + 16 = 0
2
7. Write each relation in vertex form by
completing the square.
a) y = x 2 + 6x - 3
b) y = x 2 - 4x + 5
c) y = 2x 2 + 16x + 30
d) y = - 3x 2 - 18x - 17
e) y = 2x 2 + 10x + 8
f ) y = - 3x 2 + 9x - 2
3. Solve using any strategy.
a) x 2 + 12x + 45 = 10
b) 2x 2 + 7x + 5 = 9
c) x(6x - 1) = 12
d) x(x + 3) - 20 = 5(x + 3)
8. Consider the relation y = - 4x 2 + 40x - 91.
4. Kari drew this sketch of a small suspension
bridge over a gorge near her home.
9. Martha bakes and sells her own organic dog
y
x
She determined that the bridge can be modelled by
the relation y = 0.1x 2 - 1.2x + 2. How wide is
the gorge, if 1 unit on her graph represents 1 m?
5. If a ball were thrown on Mars, its height, h, in
metres, might be modelled by the relation
h = - 1.9t 2 + 18t + 1, where t is the time in
seconds since the ball was thrown.
a) Determine when the ball would be 20 m or
higher above Mars’ surface.
b) Determine when the ball would hit the surface.
NEL
a) Complete the square to write the equation
in vertex form.
b) Determine the vertex and the equation
of the axis of symmetry.
c) Graph the relation.
treats for $15/kg. For every $1 price increase,
she will lose sales. Her revenue, R, in dollars, can
be modelled by R = - 10x 2 + 100x + 3750,
where x is the number of $1 increases. What
selling price will maximize her revenue?
10. For his costume party, Byron hung a spider from a
spring that was attached to the ceiling at one end.
Fern hit the spider so that it began to bounce up
and down. The height of the spider above the
ground, h, in centimetres, during one bounce can
be modelled by h = 10t 2 - 40t + 240, where
t seconds is the time since the spider was hit.
When was the spider closest to the ground
during this bounce?
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The Quadratic Formula
YOU WILL NEED
GOAL
• graphing calculator
Understand the development of the quadratic formula, and use
the quadratic formula to solve quadratic equations.
LEARN ABOUT the Math
Devlin says that he cannot solve the equation 2x 2 + 4x - 10 = 0 by
factoring because his graphing calculator shows him that the zeros of the
relation y = 2x 2 + 4x - 10 are not integers.
He wonders if there is a way to solve quadratic equations that cannot
be factored over the set of integers.
?
EXAMPLE 1
How can quadratic equations be solved without factoring or
using a graph?
Selecting a strategy to solve a quadratic equation
Solve 2x 2 + 4x - 10 = 0.
Kyle’s Solution: Solving a quadratic equation using the vertex form
2x 2 + 4x - 10 = 0
2(x 2 + 2x) - 10 = 0
2
= 1 and 12 = 1
2
2(x 2 + 2x + 1 - 1) - 10 = 0
336
6.4 The Quadratic Formula
Since the equation contains an x 2 term as well as an
x term, I knew that I couldn’t isolate x like I do for
linear equations. But the vertex form of a quadratic
equation does contain a single x term. I wondered
whether I could isolate x if I wrote the expression on
the left in this form. I decided to complete the square.
I factored 2 from the x 2 and x terms. I divided the
coefficient of x by 2. Then I squared my result to
determine what I needed to add and subtract
to create a perfect square within the expression
on the left side.
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6.4
2[(x 2 + 2x + 1) - 1] - 10 = 0
2[(x + 1)2 - 1] - 10 = 0
2(x + 1)2 - 2 - 10 = 0
2(x + 1)2 - 12 = 0
2(x + 1)2 = 12
12
2(x + 1)2
=
2
2
2
(x + 1) = 6
2(x + 1)2 = ; 26
x + 1 = ; 26
x = - 1 ; 26
x = - 1 + 26 and x = - 1 - 26 are the
exact solutions.
I grouped together the three terms that formed the
perfect-square trinomial and then factored.
Finally, I multiplied and combined the constants.
I isolated (x + 1)2 using inverse operations. Since
(x + 1) is squared, I took the square roots of both
sides. I remembered that there are two square
roots for every number: a positive one and a
negative one. Then I solved for x.
I got two answers. This makes sense because
these roots are the x-intercepts of the graph
of y = 2x 2 + 4x - 10.
I decided to compare the roots I calculated with
the x-intercepts of Devlin’s graph by writing these
numbers as decimals. My roots and Devlin’s
x-intercepts were the same.
The roots of 2x 2 + 4x - 10 = 0 are approximately
x = 1.4 and x = - 3.4 .
Liz’s Solution: Solving a quadratic equation by developing a formula
ax 2 + bx + c = 0
aax 2 +
b
xb + c = 0
a
b
1
b
b
b2
b 2
, 2 = * =
and a b =
a
a
2
2a
2a
4a 2
NEL
I thought I could solve for x if I could write the
standard form of an equation in vertex form, since
this form is the only one that has a single x term.
I realized that I would have to work with letters
instead of numbers, but I reasoned that the process
would be the same.
I decided to complete the square, so I factored a from
the x 2 and x terms.
To determine what I needed to add and subtract
to create a perfect square within the expression,
I divided the coefficient of x by 2. Then I squared
my result.
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b
b2
b2
b + c = 0
aax 2 + x +
a
4a 2
4a 2
b
b2
b2
b
d + c = 0
a ca x 2 + x +
a
4a 2
4a 2
b2
b 2
d + c = 0
b a ca x +
2a
4a 2
ab 2
b 2
+ c = 0
b aa x +
2a
4a 2
b 2
b2
aax +
b + c = 0
2a
4a
b2
b 2
b =
- c
2a
4a
b 2
b 2 - 4ac
aax +
b =
2a
4a
b 2
b 2 - 4ac
ax +
b =
2a
4a 2
aax +
A
ax +
b2
to the
4a2
binomial inside the brackets. I grouped
together the terms that formed the
perfect-square trinomial and then
factored. Finally, I multiplied by
a and simplified.
I added and subtracted
To solve for x, I used inverse operations.
I divided both sides by a. I took the
square root of both sides. Since the
right side represents a number, I had to
determine both the positive and
negative square roots.
b 2
b 2 - 4ac
b = ;
2a
A 4a 2
x +
; 2b 2 - 4ac
b
=
2a
2a
b
2b 2 - 4ac
x = ;
2a
2a
x =
-b ; 2b 2 - 4ac
2a
2x 2 + 4x - 10 = 0
- 4 ; 242 - 4(2)( -10)
x =
2(2)
- 4 ; 216 + 80
x =
4
- 4 ; 296
x =
4
I subtracted
b
from both sides and
2a
simplified the expression.
I reasoned that my formula could be
used for any quadratic equation.
The ; in the numerator means that
there could be two solutions: one
when you add the square root of
b 2 - 4ac to -b before dividing by
2a, and another when you subtract.
To verify my formula, I checked that
the roots were the same as Devlin’s
x-intercepts. I substituted a 2, b 4,
and c 10 into my formula.
The roots of 2x 2 + 4x - 10 = 0 are
x =
-4 - 296
- 4 + 296
and x =
.
4
4
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6.4
I calculated my roots as decimals so that
I could compare them with Devlin’s
x-intercepts. They were the same.
Reflecting
A.
Why does it make sense that a quadratic equation may have
two solutions?
B.
How are Kyle’s solution and Liz’s solution the same? How are they
different?
C.
Why does it make sense that a, b, and c are part of the quadratic
formula ?
APPLY the Math
EXAMPLE 2
Selecting a tool to verify the roots
of a quadratic equation
quadratic formula
a formula for determining the
roots of a quadratic equation of
the form ax 2 + bx + c = 0; the
quadratic formula is written using
the coefficients and the constant
in the equation:
-b ; 2b 2 - 4ac
x =
2a
Solve 5x 2 - 4x - 3 = 0. Round your solutions to two decimal places.
Verify your solutions using a graphing calculator.
Maddy’s Solution
5x 2 - 4x - 3 = 0
a = 5, b = - 4, c = - 3
- b ; 2b 2 - 4ac
x =
2a
x =
- ( - 4) ; 2(- 4)2 - 4(5)( -3)
2(5)
x =
4 ; 216 + 60
10
x =
4 ; 276
10
# 4 ; 8.718
x =
10
NEL
I noticed that the trinomial in this
equation is not factorable over
the set of integers, so I decided
to use the quadratic formula.
I identified the values of a, b,
and c.
I substituted the values for a, b,
and c and simplified.
I calculated 276 and rounded
to 3 decimal places.
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# 4 - 8.718
# 4 + 8.718
x =
or x =
10
10
x = - 0.4718
x = 1.2718
#
#
x = -0.47 and x = 1.27
Tech
Support
I knew that the ; in the formula
meant that I would have two
different solutions. I wrote the
two solutions separately.
I rounded to two decimal places.
I entered the relation
y = 5x 2 - 4x - 3 into my
graphing calculator and used the
Zero operation to verify my
solutions.
For help using a TI-83/84
graphing calculator to
determine the zeros of a
relation, see Appendix B-8.
If you are using a TI-nspire,
see Appendix B-44.
The zeros of the relation agreed with the
roots I had calculated using the formula.
EXAMPLE 3
Reasoning about solving quadratic equations
Solve each equation. Round your solutions to two decimal places.
a) 2x 2 - 10 = 8
b) 3x(5x - 4) + 2x = x 2 - 4(x - 3)
Graham’s Solution
a) 2x 2 - 10 = 8
I noticed that this quadratic equation did not contain
an x term. I reasoned that I could solve the equation
if I isolated the x 2 term.
2x 2 = 8 + 10
2x 2 = 18
I added 10 to both sides.
2x 2
18
=
2
2
I divided both sides by 2.
x2 = 9
2x 2 = ; 29
x = ;3
I took the square root of both sides.
x = 3 and x = - 3
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6.4 The Quadratic Formula
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6.4
b) 3x(5x - 4) + 2x = x 2 - 4(x - 3)
15x 2 - 12x + 2x = x 2 - 4x + 12
14x 2 - 6x - 12 = 0
a = 14, b = - 6, c = - 12
x =
-b ;
I simplified the equation by multiplying, using the
distributive property.
I rearranged the equation so that the right side was 0.
I decided to use the quadratic formula since I didn’t
quickly see the factors.
I identified the values of a, b, and c.
2b 2 - 4ac
2a
x =
- (- 6) ; 2( - 6)2 - 4(14)( -12)
2(14)
x =
6 ; 2708
28
I substituted the values of a, b, and c into the quadratic
formula and evaluated. I rounded the solutions to two
decimal places.
#
#
x = - 0.74 and x = 1.16
EXAMPLE 4
Solving a problem using a quadratic model
A rectangular field is going to be completely enclosed by 100 m of fencing.
Create a quadratic relation that shows how the area of the field will depend
on its width. Then determine the dimensions of the field that will result in
an area of 575 m2. Round your answers to the nearest hundredth of a metre.
Bruce’s Solution
Let the width of the field be w metres.
w
50 w
A = lw
w(50 - w)
50w - w 2
575 = 50w - w 2
I started with a diagram to help me organize my thinking.
I decided to represent the width of the field by w.
Since the perimeter will be 100 m, the length will have to be
100 - 2w
= 50 - w.
2
I wrote a quadratic relation for the area by multiplying the
length and the width. Then I set the area equal to 575.
0 = - w 2 + 50w - 575
I rearranged the equation so that the left side was 0.
a = - 1, b = 50, c = - 575
I decided to use the quadratic formula, so I identified
the values of a, b, and c.
w =
NEL
- b ; 2b 2 - 4ac
2a
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- 50 ; 2502 - 4(- 1)( - 575)
2( - 1)
w =
- 50 ; 22500 - 2300
-2
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I substituted these values into the formula and evaluated.
I rounded the solutions to two decimal places.
- 50 ; 2200
w =
-2
#
#
w = 17.93 and w = 32.07
The field is either 17.93 m or 32.07 m wide.
50 17.93 32.07
If w 17.93 m, the field is 32.07 m long.
50 32.07 17.93
If w 32.07 m, the field is 17.93 m long.
I used the two possible values of w to determine the
values for the length. It made sense that the roots could
be either the length or the width.
In Summary
Key Idea
• The roots of a quadratic equation of the form ax 2 + bx + c = 0 can be
determined using the quadratic formula: x =
-b ; 2b 2 - 4ac
.
2a
Need to Know
• The quadratic formula was developed by completing the square
to solve ax 2 + bx + c = 0.
• The quadratic formula provides a way to calculate the roots
of a quadratic equation without graphing or factoring.
• The solutions to the equation ax 2 + bx + c = 0 correspond to
the zeros, or x-intercepts, of the relation y = ax 2 + bx + c.
• Quadratic equations that do not contain an x term can be solved
by isolating the x 2 term.
• Quadratic equations of the form a(x - h)2 + k = 0 can be solved
by isolating the x term.
CHECK Your Understanding
1. State the values of a, b, and c that you would substitute into the
quadratic formula to solve each equation. Rearrange the equation,
if necessary.
a) x 2 + 5x - 2 = 0
c) x 2 + 6x = 0
b) 4x 2 - 3 = 0
d) 2x(x - 5) = x 2 + 1
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6.4
2. i)
Solve each equation by factoring.
ii) Solve each equation using the quadratic formula.
iii) State which strategy you prefer for each equation, and explain why.
a) x 2 + 18x - 63 = 0
b) 8x 2 - 10x - 3 = 0
3. Solve each equation.
a) 2x 2 = 50
b) x 2 - 1 = 0
c) 3x 2 - 2 = 10
d) x(x - 2) = 36 - 2x
PRACTISING
4. Determine the roots of each equation. Round the roots to two decimal
places, if necessary.
a) (x + 1)2 - 16 = 0
b) - 2(x + 5)2 + 2 = 0
c) - 3(x - 7)2 + 3 = 0
d) 4(x - 2)2 - 5 = 0
e) -6(x + 3)2 + 12 = 0
f ) 0.25(x - 4)2 - 4 = 0
5. Solve each equation using the quadratic formula.
a) 6x 2 - x - 15 = 0
b) 4x 2 - 20x + 25 = 0
c) x 2 - 16 = 0
d) 5x 2 - 11x = 0
e) x 2 + 9x + 20 = 0
f ) 12x 2 - 40 = 17x
6. Could you have solved the equations in question 5 using
a different strategy? Explain.
7. If you can solve a quadratic equation by factoring it over the set of
C
integers, what would be true about the roots you could determine
using the quadratic formula? Explain.
8. Determine the roots of each equation. Round the roots to two decimal
K
places.
a) x 2 - 4x - 1 = 0
b) 5x 2 - 6x - 2 = 0
c) 3w 2 + 8w + 2 = 0
d) 2x 2 - x - 3 = 0
e) m 2 - 5m + 3 = 0
f ) -3x 2 + 12x - 7 = 0
9. Solve each equation. Round your solutions to two decimal places.
a) 2x 2 - 5x = 3(x + 4)
b) (x + 4)2 = 2(x + 5)
c) x(x + 3) = 5 - x 2
d) 3x(x + 4) = (4x - 1)2
e) (x - 2)(2x + 3) = x + 1
f ) (x - 3)2 + 5 = 3(x + 1)
10. Solve each equation. Round your solutions to two decimal places.
a) 2x 2 + 5x - 14 = 0
b) 3x 2 + 7.5x = 21
c) 3x(0.4x + 1) = 8.4
d) 0.2x 2 = - 0.5x + 1.4
11. a) What do you notice about your solutions in question 10?
b) How could you have predicted this before using the quadratic
formula?
NEL
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12. Algebraically determine the points of intersection of the parabolas
T
y = 2x 2 + 5x - 8 and y = - 3x 2 + 8x - 1.
13. Calculate the value of x.
x7
x
x9
14. A trained stunt diver is diving off a platform that is 15 m high into
A
a pool of water that is 45 cm deep. The height, h, in metres, of the
stunt diver above the water is modelled by h = - 4.9t 2 + 1.2t + 15,
where t is the time in seconds after starting the dive.
a) How long is the stunt diver above 15 m?
b) How long is the stunt diver in the air?
15. A rectangle is 5 cm longer than it is wide. The diagonal of the
rectangle is 18 cm. Determine the dimensions of the rectangle.
16. A square lawn is surrounded by a concrete walkway that is 2.0 m wide,
as shown at the left. If the area of the walkway equals the area of the
lawn, what are the dimensions of the lawn? Express the dimensions
to the nearest tenth of a metre.
2.0 m
17. Use a chart like the one below to compare the advantages of solving
a quadratic equation by factoring and by using the quadratic formula.
Provide an example of an equation that you would solve using each
strategy.
Strategy
Advantages
Example
Factoring
Quadratic
Formula
Extending
18. Determine the points of intersection of the line y = 2x + 5 and
the circle x 2 + y 2 = 36.
19. Determine a quadratic equation, in standard form, that has each pair
of roots.
a) x = - 3 and x = 5
b) x =
2 ; 25
3
20. Three sides of a right triangle are consecutive even numbers, when
measured in centimetres. Calculate the length of each side.
344
6.4 The Quadratic Formula
NEL
6.5
Interpreting Quadratic
Equation Roots
GOAL
Determine the number of roots of a quadratic equation, and relate
these roots to the corresponding relation.
YOU WILL NEED
• graphing calculator
• grid paper
• ruler
INVESTIGATE the Math
Quadratic relations may have two, one, or no x-intercepts.
y
8
y x2 x 6
6
4
2
This graph shows that the
quadratic equation
- x 2 + x + 6 = 0 has
two solutions, x = - 2
and x = 3 .
x
-4 -2
0
2
-2
4
6
y
8
This graph shows that
the quadratic equation
x 2 - 6x + 9 = 0 has
one solution, x = 3 .
6
4
2
x
-4 -2
0
-2
2 4 6
y x2 6x 9
y
8
This graph shows that
the quadratic equation
2x 2 - 4x + 5 = 0 has
no solutions.
6
4
2
2 y 2x 4x 5
x
-4 -2
?
NEL
0
-2
2
4
6
How can you determine the number of solutions to a quadratic
equation without solving it?
Chapter 6
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Quadratic Relation
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Copy and complete this table using a graphing calculator. The first line
has been completed for you.
Sketch of Graph
Quadratic Equation
Used to Determine
x-intercepts
y
2x 2 - 12x + 13 = 0
y = 2x 2 - 12x + 13
2
x
-2
0
-2
2
4
-4
-6
Roots of the Equation
- b ; 2b 2 - 4ac
2a
- (- 12) ; 2( - 12)2 - 4(2)(13)
x =
2(2)
12 ; 240
x =
4
x 1.42 or x 4.58
x =
y = - 2x 2 - 4x - 2
y = - 3x 2 + 9x + 12
y = x 2 - 6x + 13
y = - 2x 2 - 4x - 5
y = x 2 + 6x + 9
How is the number of real number solutions related to the value
of the discriminant?
real numbers
B.
the set of numbers that
corresponds to each point on the
number line shown; fractions,
decimals, integers, and numbers
like 22 are all real numbers
Reflecting
C.
Why are there no real number solutions when the discriminant
is negative?
1
D.
Why is there one real number solution when the discriminant is zero?
E.
Why are there two real number solutions when the discriminant
is positive?
0
1
discriminant
the expression b 2 - 4ac in the
quadratic formula
APPLY the Math
EXAMPLE 1
Connecting the real roots to the x-intercepts
Without solving, determine the number of real roots of each equation and
describe the graph of the corresponding relation.
a) 3x 2 + 4x + 5 = 0
b) - 2x 2 + 7x + 1 = 0
c) 9x 2 - 12x + 4 = 0
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6.5
Steve’s Solution
a) 3x 2 + 4x + 5 = 0
D = b 2 - 4ac
= 42 - 4(3)(5)
= 16 - 60
= - 44
There are no real roots.
I substituted the values of a,
b, and c into the discriminant,
which I called D.
Since the discriminant is
negative, there are no
real roots.
The graph of y = 3x 2 + 4x + 5 has
no x-intercepts. Since a 7 0, the
parabola opens upward and its vertex
is above the x-axis.
b) - 2x 2 + 7x + 1 = 0
D = b 2 - 4ac
= 72 - 4(- 2)(1)
= 49 + 8
= 57
There are two real roots.
I substituted the values of a,
b, and c into the discriminant.
Since the discriminant is
positive, there are two
real roots.
The graph of y = - 2x 2 + 7x + 1
has two x-intercepts. Since a 6 0,
the parabola opens downward and
its vertex is above the x-axis.
c) 9x 2 - 12x + 4 = 0
D = b 2 - 4ac
= ( - 12)2 - 4(9)(4)
= 144 - 144
= 0
I substituted the values of a,
b, and c into the discriminant.
There is one real root.
Since the discriminant is zero,
there is one real root.
The graph of y = 9x 2 - 12x + 4
has one x-intercept. Since a 7 0,
the parabola opens upward and its
vertex is on the x-axis.
NEL
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EXAMPLE 2
Selecting a strategy to determine
the number of zeros
Determine the number of zeros for y = - 2x 2 + 16x - 35.
John’s Solution: Completing the square
y = - 2x 2 + 16x - 35
y = - 2(x 2 - 8x) - 35
y = - 2(x 2 - 8x + 16 - 16) - 35
y = - 2(x - 4)2 + 32 - 35
y = - 2(x - 4)2 - 3
I completed the square to
determine the vertex of the
parabola. The vertex is at (4, -3).
Since a 6 0, the parabola opens
downward.
There are no zeros.
Since the vertex is below the
x-axis and the parabola opens
downward, I knew that it could
never cross the x-axis.
Erin’s Solution: Using the quadratic formula
y = - 2x 2 + 16x - 35
0 = - 2x 2 + 16x - 35
x =
-16 ; 2162 - 4( -2)( -35)
2(-2)
x =
-16 ; 2-24
- 4
There are no real number solutions
to the equation, so the relation has
no x-intercepts.
348
6.5 Interpreting Quadratic Equation Roots
The zeros occur when y = 0,
so I substituted y = 0 into the
relation. This resulted in a
quadratic equation.
I substituted a = - 2, b = 16,
and c = - 35 into the quadratic
formula.
I tried to calculate the square
root of -24, but my calculator
displayed this error message.
My calculator displayed an error
message because there is no real
number that can be multiplied
by itself to give a negative result.
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6.5
Cathy’s Solution: Using the discriminant
y = - 2x 2 + 16x - 35
D = 162 - 4(- 2)( - 35)
= 256 - 280
= - 24
I substituted a = - 2, b = 16,
and c = - 35 into the
discriminant and evaluated.
Since the discriminant is
negative, there are no zeros.
The relation has no zeros.
In Summary
Key Idea
• You can use the quadratic formula to determine whether a quadratic
equation has two, one, or no real solutions, without solving the
equation.
Need to Know
• The value of the expression D = b 2 - 4ac gives the number of real
solutions to a quadratic equation and the number of zeros in the graph
of its corresponding relation.
• If D 7 0, there are two distinct real roots, or zeros.
• If D = 0, there is one real root, or zero.
• If D 6 0, there are no real roots, or zeros.
• The direction of opening of a graph and the position of the vertex
determines whether the graph has two, one, or no zeros and indicates
whether the corresponding equation has two, one, or no real roots.
Two Real Roots
y
One Real Root
No Real Roots
y
y
x
x
x
CHECK Your Understanding
1. a) Determine the roots of x 2 - 6x + 5 = 0 by using
the quadratic formula and by factoring.
b) What do your results for part a) tell you about the graph
of y = x 2 - 6x + 5?
c) Verify your answer for part b) using the discriminant.
NEL
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2. Determine the number of real solutions that each equation has,
without solving the equation. Explain your reasoning.
a) (x - 1)2 + 3 = 0
b) -2(x - 5)2 + 8 = 0
c) 5(x + 3)2 = 0
PRACTISING
3. Use the discriminant to determine the number of real solutions
that each equation has.
a) x 2 + 3x - 5 = 0
b) 6x 2 + 5x + 12 = 0
c) -x 2 + 8x = 12
d) -2x 2 + 8x - 8 = 0
e) 3x 2 + 2x = 5x + 12
f ) -17x - 9 = 4x 2 - 5x
4. State the number of times that each relation passes through the x-axis.
Justify your answer.
a) y = 3(x + 2)2 - 5
b) y = - 2(x + 5)2 - 8
c) y = 2(x - 7)2
d) y = 5(x - 12)2 + 81
e) y = - 4.9x 2 + 5
f ) y = - 6x 2
5. Without graphing, determine the number of zeros that each relation has.
K
a) y = 3(x + 2)2 + 5
b) y = - 2x 2 + 3x - 7
c) y = x(x - 7)
d) y = 5(x + 2)(x + 2)
e) y = 3x 2 + 6x - 8
f ) y = - 6x 2 + 9
6. Emma sells her handmade jewellery at a local market. She has always
A
sold her silver toe rings for $10 each, but she is thinking about
raising the price. Emma knows that her weekly revenue, r, in dollars,
is modelled by r = 250 + 5n - 2n 2, where n is the amount that
she increases the price. Is it possible for Emma to earn $500 in
revenue? Explain.
7. In some places, a suspension bridge is the only passage over a river.
The height of one such bridge, h, in metres, above the riverbed can be
modelled by h = 0.005x 2 + 24.
a) How many zeros do you expect the relation to have? Why?
b) If the area was flooded, how high could the water level rise before
the bridge was no longer safe to use?
8. The height of a super ball, h, in metres, can be modelled by
h = - 4.9t 2 + 10.78t + 1.071, where t is the time in seconds since
the ball was thrown.
a) How many zeros do you expect this relation to have? Why?
b) Verify your answer for part a) algebraically.
c) How many times do you think the ball will pass through a height
of 5 m? 7 m? 9 m?
d) Verify your answers for part c) algebraically.
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6.5 Interpreting Quadratic Equation Roots
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6.5
9. Determine whether the vertex of each parabola lies above, below, or on
the x-axis. Explain how you know.
a) h = 2t 2 - 4t + 1.5
c) h = 5t 2 - 30t + 45
b) h = 0.5t 2 - 2t + 0.5
d) h = 0.5t 2 - 4t + 7.75
10. For what value(s) of k does the equation y = 5x 2 + 6x + k have each
number of roots?
a) two real roots
b) one real root
c) no real roots
11. Meg went bungee jumping from the Bloukrans River bridge in South
Africa last summer. During the free fall on her first jump, her height
above the water, h, in metres, was modelled by h = - 5t 2 + t + 216,
where t is the time in seconds since she jumped.
a) How high above the water is the platform from which she jumped?
b) Show that if her hair just touches the water on her first jump, the
corresponding quadratic equation has two solutions. Explain what
the solutions mean.
12. In the relation y = 4x 2 + 24x - 5, for which values of y will the
corresponding equation have no solutions?
13. A tangent is a line that touches a circle at exactly one point. For what
T
values of k will the line y = x + k be tangent to the circle
x 2 + y 2 = 25?
14. Sasha claims that the discriminant of a quadratic relation will never be
C
negative if the relation can be written in the form y = a(x - r)(x - s).
Do you agree or disagree? Explain.
15. a) Create a quadratic relation, in vertex form, that has two zeros.
Then write your relation in standard form. Use the discriminant
to verify that it has two zeros.
b) Create a quadratic relation, in vertex form, that has one zero. Then
write your relation in standard form. Use the discriminant to verify
that it has one zero.
c) Create a quadratic relation, in vertex form, that has no zeros. Then
write your relation in standard form. Use the discriminant to verify
that it has one zero.
Safety Connection
Bungee jumping is an activity
associated with a high degree
of risk. This activity should
only be performed under
the direction of trained
professionals.
Extending
16. a) Write three quadratic equations in factored form.
b) Expand and simplify your equations.
c) Determine the discriminant for each equation.
d) Explain how you can use the discriminant to determine whether
an equation is factorable.
17. Determine the number of points of intersection of the relations
y = (x + 3)2 and y = - 2x 2 - 5.
NEL
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Solving Problems Using
Quadratic Models
YOU WILL NEED
GOAL
• grid paper
• ruler
• graphing calculator
Solve problems that can be modelled by quadratic relations using
a variety of tools and strategies.
LEARN ABOUT the Math
The volunteers at a food bank are arranging a concert to raise money.
They have to pay a set fee to the musicians, plus an additional fee
to the concert hall for each person attending the concert. The relation
P = - n 2 + 580n - 48 000 models the profit, P, in dollars, for the
concert, where n is the number of tickets sold.
?
EXAMPLE 1
How can you determine the number of tickets that must be sold
to break even and to maximize the profit?
Selecting a strategy to analyze a quadratic relation
Calculate the number of tickets they must sell to break even. Determine
the number of tickets they must sell to maximize the profit.
Jack’s Solution: Completing the square
P = - n 2 + 580n - 48 000
P = - (n 2 - 580n) - 48 000
P = - (n 2 - 580n + 84 100 - 84 100) - 48 000
P = - [(n - 290)2 - 84 100] - 48 000
P = - (n - 290)2 + 84 100 - 48 000
P = - (n - 290)2 + 36 100
I completed the square to write the
relation in vertex form so I could
determine the maximum profit first.
The volunteers must sell 290 tickets to earn a
maximum profit of $36 100 for the food bank.
Since a 6 0, the parabola opens
downward. The y-coordinate of the vertex
(290, 36 100) is the maximum value.
0 = - (n - 290)2 + 36 100
I set P = 0 to calculate the break-even
points.
(n - 290)2 = 36 100
2(n - 290)2 = ; 236 100
n - 290 = ; 190
n = 290 + 190 or n = 290 - 190
n = 480
n = 100
I used inverse operations to solve for n.
The volunteers break even if they sell 480 tickets
or 100 tickets.
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6.6 Solving Problems Using Quadratic Models
NEL
6.6
Dineke’s Solution: Factoring the relation
P = - n 2 + 580n - 48 000
P = - (n 2 - 580n + 48 000)
P = - (n - 480)(n - 100)
I factored the relation to determine the
break-even points first.
0 = - (n - 480)(n - 100)
0 = n - 480 or 0 = n - 100
n = 480
n = 100
I knew that the break-even points would
occur when the profit equalled zero, so I
set P ⫽ 0. Then I used inverse operations
to solve for n.
The volunteers break even if they sell 480 tickets
or 100 tickets.
Since the zeros are the same distance
from the axis of symmetry, I used them
to determine the equation of the axis of
symmetry.
480 + 100
n =
2
n = 290
P = - (n - 480)(n - 100)
P = - (290 - 480)(290 - 100)
P = - (- 190)(190)
P = 36 100
Therefore, the volunteers will earn a maximum profit
of $36 100 for the food bank if they sell 290 tickets.
The equation of the axis of symmetry,
n ⫽ 290, gave the n-coordinate of the
vertex. I substituted n ⫽ 290 into the
equation to determine the P-coordinate
of the vertex.
Reflecting
A.
How are Jack’s solution and Dineke’s solution the same? How are they
different?
B.
Will both strategies always work? Why or why not?
C.
Whose strategy would you have used for this problem? Explain your
choice.
APPLY the Math
EXAMPLE 2
Solving a problem by creating a quadratic model
Alexandre was practising his 10 m platform dive. Because of gravity, the
relation between his height, h, in metres, and the time, t, in seconds, after
he dives is quadratic. If Alexandre reached a maximum height of 11.225 m
after 0.5 s, how long was he above the water after he dove?
NEL
Chapter 6
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Burns’s Solution
h
(0.5, 11.225)
I decided to start with a sketch that
included the given information. The
y-intercept is the starting position.
The maximum height and time are
the coordinates of the vertex,
(time, maximum height).
10
(0, 10)
5
t
y = a(x - h)2 + k
10 = a(0 - 0.5)2 + 11.225
10 = a( - 0.5)2 + 11.225
10 - 11.225 = 0.25a
0.25a
- 1.225
=
0.25
0.25
-4.9 = a
h = - 4.9(t - 0.5)2 + 11.225
0 = - 4.9(t - 0.5)2 + 11.225
- 11.225
- 4.9(t - 0.5)2
=
- 4.9
- 4.9
2.291 (t - 0.5)2
; 22.291 = 2(t - 0.5)2
; 1.514 t - 0.5
0.5 ; 1.514 t
0.5 + 1.514 t or 0.5 - 1.514 t
2.01 t
- 1.01 t
Alexandre was above the water for
about 2.0 s after he dove.
354
6.6 Solving Problems Using Quadratic Models
Since I knew the coordinates of the
vertex, I determined a model for the
height of the diver in vertex form.
Substituting the vertex given for (h, k)
and the initial height of the diver for a
point (x, y), I used inverse operations to
solve for a.
Since I was determining when
Alexandre hit the water, I set h 0.
I used inverse operations to solve for t.
The answer 1.01 didn’t make sense
since time cannot be negative in this
situation, so I didn’t use it.
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6.6
Reasoning to determine an appropriate solution
Statisticians use various models to make predictions about population
growth. Ontario’s population (in 100 000s) can be modelled by the
relation P = 0.007x 2 + 0.196x + 21.5, where x is the number
of years since 1900.
a) Using this model, what was Ontario’s population in 1925?
b) When will Ontario’s population reach 15 million?
Ontario’s Population since 1900
y
120
Population (100 000s)
EXAMPLE 3
100
80
60
40
20 P 0 007x2 0 196x 21 5
x
0
20 40 60 80 100 120
Number of years since 1900
Blair’s Solution
a) x = 1925 - 1900
x = 25
P = 0.007x 2 + 0.196x + 21.5
P = 0.007(25)2 + 0.196(25) + 21.5
P = 30.775
The population was about 3 077 500 in 1925.
P = 0.007x 2 + 0.196x + 21.5
150 = 0.007x 2 + 0.196x + 21.5
b)
150 - 150 = 0.007x 2 + 0.196x + 21.5 - 150
0 = 0.007x 2 + 0.196x - 128.5
Using x as the number of years since 1900, I
subtracted 1900 from 1925. I substituted my result
into the relation to determine the population in 1925.
The relation gave the population in 100 000s, so
I multiplied my answer by 100 000.
The population was given in 100 000s, and
15 million 150 100 000. So, I used 150 for P.
I rearranged my equation so that I could use
the quadratic formula to determine its roots.
a 0.007, b 0.196, c 128.5
x =
- b; 2b 2 - 4ac
2a
x =
- 0.196 ; 20.1962 - 4(0.007)( -128.5)
2(0.007)
- 0.196 ; 23.636
0.014
x 150.20 or x 122.20
x =
Ontario’s population will be
about 15 000 000 in 2022.
NEL
I thought x = - 150.21 made no sense, since this
would mean that the population was 15 000 000 in
1750, which I know is not reasonable. So, I used
the other answer and added it to 1900.
Chapter 6
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Solving a problem by creating a quadratic model
Lila is creating dog runs for her dog kennel. She can afford 30 m
of chain-link fence to surround four dog runs. The runs will be
attached to a wall, as shown in the diagram. To achieve the
maximum area, what dimensions should Lila use for each run
and for the combined enclosure?
Mitsu’s Solution
I started by sketching the dog runs. I let w represent
the width of each run. I let l represent the total length
of the enclosure.
w
w
w
w
w
To express the length in terms of the width, I subtracted
the amount of fencing needed for the 5 fence sides
that are perpendicular to the wall from the amount
of fencing Lila can afford.
l = 30 – 5w
A = l * w
A = (30 - 5w) * w
A = 30w - 5w 2
A = - 5w 2 + 30w
A = - 5(w 2 - 6w)
A = - 5(w 2 - 6w + 9 - 9)
A = - 5[(w - 3)2 - 9]
A = - 5(w - 3)2 - (- 5)9
A = - 5(w - 3)2 + 45
(3, 45) is the vertex, so the maximum area
is 45 m2. It occurs when the width of each
run is 3 m.
I wrote an equation for the area and simplified it.
Since I wanted to maximize the area, I completed
the square to determine the vertex.
l = 30 - 5w
= 30 - 5(3)
= 15
I substituted the width into my expression for the
length to determine the length of the combined
enclosure.
The dimensions of the combined enclosure
should be 15 m by 3 m, and the dimensions of
each run should be 3 m by 3.75 m.
15 4 3.75
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6.6
In Summary
Key Idea
• When solving a problem that involves a quadratic relation, follow these
suggestions:
• Write the relation in the form that is most helpful for the given
situation.
• Use the vertex form to determine the maximum or minimum value
of the relation.
• Use the standard form or factored form to determine the value
of x that corresponds to a given y-value of the relation. You
may need to use the quadratic formula.
Need to Know
• A problem may have only one realistic solution, even when the
quadratic equation that is used to represent the problem has two real
solutions. When you solve a quadratic equation, check to make sure
that your solutions make sense in the context of the situation.
CHECK Your Understanding
1. For each relation, explain what each coordinate of the vertex represents
and what the zeros represent.
a) a relation that models the height, h, of a ball that has been kicked
from the ground after time t
b) a relation that models the height, h, of a ball when it is a distance
of x metres from where it was thrown from a second-floor balcony
c) a relation that models the profit earned, P, on an item at a given
selling price, s
d) a relation that models the cost, C, to create n items using a piece
of machinery
e) a relation that models the height, h, of a swing above the ground
during one swing, t seconds after the swing begins to move forward
For questions 2 to 17, round all answers to two decimal places, where necessary.
2. A model rocket is shot straight up from the roof of a school. The
height, h, in metres, after t seconds can be approximated by
h = 15 + 22t - 5t 2.
a) What is the height of the school?
b) How long does it take for the rocket to pass a window that is 10 m
above the ground?
c) When does the rocket hit the ground?
d) What is the maximum height of the rocket?
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PRACTISING
3. Water from a fire hose is sprayed on a fire that is coming from a
window. The window is 15 m up the side of a wall. The equation
H = - 0.011x 2 + 0.99x + 1.6 models the height of the jet of water,
H, and the horizontal distance it can travel from the nozzle, x, both
in metres.
a) What is the maximum height that the water can reach?
b) How far back could a firefighter stand, but still have the water
reach the window?
4. Brett is jumping on a trampoline in his backyard. Each jump takes
about 2 s from beginning to end. He passes his bedroom window,
which is 4 m high, 0.4 s into each jump. By modelling Brett’s height
with a quadratic relation, determine his maximum height.
5. Pauline wants to sell stainless steel water bottles as a school fundraiser.
She knows that she will maximize profits, and raise $1024, if she sells
the bottles for $28 each. She also knows that she will lose $4160 if she
sells the bottles for only $10 each.
a) Write a quadratic relation to model her profit, P, in dollars, if she
sells the bottles for x dollars each.
b) What selling price will ensure that she breaks even?
Safety Connection
Smoke detectors provide early
warning of fire or smoke.
Monitored smoke detectors
send a signal to a monitoring
station.
6. A professional stunt performer at a theme park dives off a tower, which
K
is 21 m high, into water below. The performer’s height, h, in metres,
above the water at t seconds after starting the jump is given by
h 4.9t 2 21.
a) How long does the performer take to reach the halfway point?
b) How long does the performer take to reach the water?
c) Compare the times for parts a) and b). Explain why the time
at the bottom is not twice the time at the halfway point.
7. Harold wants to build five identical pigpens, side by side, on his farm
A
using 30 m of fencing. Determine the dimensions of the enclosure that
would give his pigs the largest possible area. Calculate this area.
8. A biologist predicts that the deer population, P, in a certain national
park can be modelled by P = 8x 2 - 112x + 570, where x is the
number of years since 1999.
a) According to this model, how many deer were in the park in 1999?
b) In which year was the deer population a minimum? How many
deer were in the park when their population was a minimum?
c) Will the deer population ever reach zero, according to this model?
d) Would you use this model to predict the number of deer
in the park in 2020? Explain.
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6.6
9. The depth underwater, d, in metres, of Daisy the dolphin during a dive can
be modelled by d = 0.1t 2 - 3.5t + 6, where t is the time in seconds
after the dolphin begins her descent toward the water.
a) How long was Daisy underwater?
b) How deep did Daisy dive?
10. The cost, C, in dollars per hour, to run a machine can be modelled by
C = 0.01x 2 - 1.5x + 93.25, where x is the number of items
produced per hour.
a) How many items should be produced each hour to minimize the cost?
b) What production rate will keep the cost below $53?
11. Nick has a beautiful rectangular garden, which measures 3 m by 3 m.
He wants to create a uniform border of river rocks around three sides
of his garden. If he wants the area of the border and the area of his
garden to be equal, how wide should the border be?
12. A ball was thrown from the top of a playground jungle gym, which is
1.5 m high. The ball reached a maximum height of 4.2 m when it was
3 m from where it was thrown. How far from the jungle gym was the
ball when it hit the ground?
13. The sum of the squares of three consecutive even integers is 980.
T
Determine the integers.
14. Maggie can kick a football 34 m, reaching a maximum height of 16 m.
a) Write an equation to model this situation.
b) To score a field goal, the ball has to pass between the vertical poles
and over the horizontal bar, which is 3.3 m above the ground.
How far away from the uprights can Maggie be standing so that
she has a chance to score a field goal?
15. a) Create a problem that you could model using a quadratic relation
C
and you could solve using the corresponding quadratic equation.
b) Create a problem that you could model using a quadratic relation
and you could solve by determining the coordinates of the vertex.
Extending
16. Mark is designing a pentagonal-shaped play area for a daycare facility.
He has 30 m of nylon mesh to enclose the play area. The triangle in
the diagram is equilateral. Calculate the dimensions of the rectangle
and the triangle, to the nearest tenth of a metre, that will maximize
the area he can enclose for the play area.
17. Richie walked 15 m diagonally across a rectangular field. He
then returned to his starting position along the outside of the field.
The total distance he walked was 36 m. What are the dimensions
of the field?
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Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
• See Lesson 6.4, Example 1.
• Try Chapter Review
Q:
How can you solve a quadratic equation that is not
factorable over the set of integers, without graphing?
A:
If the quadratic equation is in the form ax 2 + bx + c = 0, you can
Questions 8 to 10.
use the quadratic formula: x =
-b ; 2b 2 - 4ac
.
2a
EXAMPLE
Solve 3x 2 - 7x - 5 = 0. Round to two decimal places.
Solution
x =
- b ; 2b 2 - 4ac
, where a = 3, b = - 7, and c = - 5
2a
x =
-(-7) ; 2(- 7)2 - 4(3)( -5)
2(3)
x =
7 ; 249 + 60
6
x =
7 ; 2109
6
7 + 2109
7 - 2109
or x =
x =
6
6
#
#
x = 2.91
x = -0.57
Study
Aid
Q:
How can you use part of the quadratic formula to determine
the number of real solutions that a quadratic equation has?
A:
You can use the discriminant, D = b 2 - 4ac. If D 6 0, there are no
real solutions. If D = 0, there is one real solution. If D 7 0, there
are two real solutions.
Q:
When using a quadratic model, how do you decide whether
you should determine the vertex or solve the corresponding
equation?
A:
If you want to determine a maximum or minimum value, then you
should locate the vertex of the relation. If you are given a specific
value of y (any number, including 0), then you should solve the
corresponding equation.
• See Lesson 6.5, Examples 1
and 2.
• Try Chapter Review
Questions 11 and 12.
Study
Aid
• See Lesson 6.6, Examples 1
and 2.
• Try Chapter Review
Questions 13 to 18.
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Chapter Review
PRACTICE Questions
Lesson 6.1
1. Solve each equation.
a) (2x - 5)(3x + 8) = 0
b) x 2 + 12x + 32 = 0
c) 3x 2 - 10x - 8 = 0
d) 3x 2 - 5x + 5 = 2x 2 + 4x - 3
e) 2x 2 + 5x - 1 = 0
f ) 5x(x - 1) + 5 = 7 + x(1 - 2x)
2. The safe stopping distance, in metres, for a
boat that is travelling at v kilometres per hour
in calm water can be modelled by the relation
d = 0.002(2v 2 + 10v + 3000).
a) What is the safe stopping distance
if the boat is travelling at 12 km/h?
b) What is the initial speed of the boat
if it takes 15 m to stop?
Lesson 6.2
3. Determine the value of c needed to create
a perfect-square trinomial.
a) x 2 + 8x + c
b) x 2 - 16x + c
c) x 2 + 19x + c
d) 2x 2 + 12x + c
e) - 3x 2 + 15x + c
f ) 0.1x 2 - 7x + c
Lesson 6.3
6. A basketball player makes a long pass to another
player. The path of the ball can be modelled by
y = - 0.2x 2 + 2.4x + 2, where x is the
horizontal distance from the player and y is the
height of the ball above the court, both in metres.
Determine the maximum height of the ball.
7. Cam has 46 m of fencing to enclose a meditation
space on the grounds of his local hospital. He has
decided that the meditation space should be
rectangular, with fencing on only three sides.
What dimensions will give the patients the
maximum amount of meditation space?
Lesson 6.4
8. Solve each equation.
a) 3x 2 - 4x - 10 = 0
b) -4x 2 + 1 = - 15
c) x 2 = 6x + 10
d) (x - 3)2 - 4 = 0
e) (2x + 5)(3x - 2) = (x + 1)
f ) 1.5x 2 - 6.1x + 1.1 = 0
9. The height, h, in metres, of a water balloon that
is launched across a football stadium can be
modelled by h = - 0.1x 2 + 2.4x + 8.1, where
x is the horizontal distance from the launching
position, in metres. How far has the balloon
travelled when it is 10 m above the ground?
4. Complete the square to write each quadratic
relation in vertex form.
a) y = x 2 + 8x - 2
b) y = x 2 - 20x + 95
c) y = - 3x 2 + 12x - 2
d) y = 0.2x 2 - 0.4x + 1
e) y = 2x 2 + 10x - 12
f ) y = - 4.9x 2 - 19.6x + 12
5. Consider the relation y = - 3x 2 - 12x - 2.
a) Write the relation in vertex form
by completing the square.
b) State the transformations that must be applied
to y = x 2 to draw the graph of the relation.
c) Graph the relation.
NEL
10. A chain is hanging between two posts so that its
height above the ground, h, in centimetres, can be
determined by h = 0.0025x 2 - 0.9x + 120,
where x is the horizontal distance from one post,
in centimetres. How far from the post is the
chain when it is 50 cm from the ground?
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Lesson 6.5
15. Tickets to a school dance cost $5, and the
11. Without solving, determine the number
of solutions that each equation has.
a) 2x 2 - 5x + 1 = 0
b) - 3.5x 2 - 2.1x - 1 = 0
c) x 2 + 5x + 8 = 0
d) 4x 2 - 15 = 0
e) 5(x 2 + 2x + 5) = - 2(2x - 25)
12. Without graphing, determine the number of
x-intercepts that each relation has.
a) y = (x - 4)(2x + 9)
b) y = - 1.8(x - 3)2 + 2
c) y = 2x 2 + 8x + 14
d) y = 2x(x - 5) + 7
e) y = - 1.4x 2 - 4x - 5.4
Lesson 6.6
13. Skydivers jump out of an airplane at an altitude
of 3.5 km. The equation H = 3500 - 5t 2
models the altitude, H, in metres, of the
skydivers at t seconds after jumping out
of the airplane.
a) How far have the skydivers fallen after 10 s?
b) The skydivers open their parachutes at an
altitude of 1000 m. How long did they
free fall?
projected attendance is 300 people. For every
$0.50 increase in the ticket price, the dance
committee projects that attendance will decrease
by 20. What ticket price will generate $1562.50
in revenue?
16. A room has dimensions of 5 m by 8 m. A rug
3
of the floor and leaves a uniform strip
4
of the floor exposed. How wide is the strip?
covers
17. Two integers differ by 12 and the sum of their
squares is 1040. Determine the integers.
18. The student council at City High School is
thinking about selling T-shirts. To help them
decide what to do, they conducted a school-wide
survey. Students were asked, “Would you buy
a school T-shirt at this price?” The results of
the survey are shown.
T-Shirt
Price, t ($)
Students Who
Would Buy, N
4.00
923
6.00
752
8.00
608
14. The arch of the Tyne bridge in England is modelled
10.00
455
- 0.008x 2 - 1.296x + 107.5, where h
12.00
287
by h =
is the height of the arch above the riverbank and
x is the horizontal distance from the riverbank,
both in metres. Determine the height of the arch.
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Chapter Review
Revenue, R ($)
a) Use the table to determine the revenue
for each possible price.
b) Draw a scatter plot relating the revenue, R,
to the T-shirt price, t. Sketch a curve of
good fit.
c) Verify that the number of students, N,
who would buy a T-shirt for t dollars
can be approximated by the relation
N = 1230 - 78t.
d) Use the equation in part c) to create
an algebraic expression for the revenue.
e) The student council needs to bring in
revenue of at least $4750. What price
range can they consider?
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Chapter Self-Test
Round all answers to two decimal places where necessary.
1. Use the graph of y = 3x 2 + 6x - 7 at the right to estimate
the solutions to each equation.
a) 3x 2 + 6x - 7 = 0
b) 3x 2 + 6x - 7 = - 7
c) 3x 2 + 6x - 9 = 0
x
-4
c) 2x 2 - 8 = 24
d) 2(x - 1)2 - 5 = 0
3. Complete the square to determine the vertex of each parabola.
a) y = 2x 2 + 12x - 14
-2
0
-2
2
-4
2. Determine the roots of each equation.
a) x 2 + 5x - 14 = 0
b) 5x 2 - 9x + 1 = 0
y
2
b) y = 3x 2 - 15x - 24
-6
-8
-10
y 3x2 6x 7
4. Can all quadratic relations be written in vertex form by completing
the square? Justify your answer.
5. Without solving, determine the number of real roots that each relation
has. Justify your answers.
a) y = 2x 2 - 4x + 7 b) y = 3(x - 4)(x - 4) c) y = (x - 3)2
6. April sells specialty teddy bears at various summer festivals. Her profit for
a week, P, in dollars, can be modelled by P = - 0.1n 2 + 30n - 1200,
where n is the number of teddy bears she sells during the week.
a) According to this model, could April ever earn a profit of $2000
in one week? Explain.
b) How many teddy bears would she have to sell to break even?
c) How many teddy bears would she have to sell to earn $500?
d) How many teddy bears would she have to sell to maximize
her profit?
7. Serge and Francine have 24 m of fencing to enclose a vegetable garden
at the back of their house. Determine the dimensions of the largest
rectangular garden they could enclose, using the back of their house
as one of the sides of the rectangle.
Process
Checklist
✔ Question 1: Did you
compare the algebraic and
graphical representations
to help you estimate?
✔ Questions 4 and 5: Did you
communicate using
correct mathematical
vocabulary as you justified
your answers?
✔ Question 6: Did you make
connections between the
quadratic equation and the
situation that it is
modelling?
8. Give two reasons why 3x 2 + 6x + 6 cannot be a perfect square.
9. A rapid-transit company has 5000 passengers daily, each currently
paying a $2.25 fare. For each $0.50 increase, the company estimates
that it will lose 150 passengers daily. If the company must be paid at
least $15 275 each day to stay in business, what minimum fare must
they charge to produce this amount of revenue?
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Chapter Task
Up and Over
On Earth, the quadratic relation h = - 5t 2 + ut + h 0 can be used
to determine the height of an object that has been thrown as it travels
through the air, measured from a reference point. In this relation, h is the
height of the object in metres, t is the time in seconds since the object was
thrown, u is the initial velocity, and h 0 is the initial height.
11 m
6m
Task
Myrtle throws a ball upward from a second-floor balcony, 6 m above
the ground, with an initial velocity of 2 m/s. In this situation, u = 2
and h 0 = 6, so the relation that models the height of the ball is
h = - 5t 2 + 2t + 6. Myrtle knows that changing the velocity with
which she throws the ball will change the maximum height of the ball.
Myrtle wants to know with what velocity she must throw the ball
to make it pass over a tree that is 11 m tall.
?
What initial velocity will result in a maximum height of 11 m?
A.
Suppose that Myrtle just dropped the ball from the balcony, with an
initial velocity of 0 m/s. Write a quadratic relation to model this situation.
B.
What is the maximum height of the ball in part A?
C.
Complete the square of h = - 5t 2 + 2t + 6 to determine
the maximum height of the ball when Myrtle throws the ball
with an initial velocity of 2 m/s.
D.
Will Myrtle have to increase or decrease the initial velocity with which
she throws the ball for it to clear the tree? Explain how you know.
E.
Create relations to model the height of the ball when it is thrown from
a second-floor balcony with initial velocities of 4 m/s and 6 m/s. Then
determine the maximum height of the ball for each relation.
F.
Create a scatter plot to show the maximum heights for initial velocities
of 0 m/s, 2 m/s, 4 m/s, and 6 m/s. Is this relation quadratic? Explain
how you know.
G.
Use quadratic regression to determine an algebraic model
for your graph for part F.
H.
Use the model you created for part G to determine the initial velocity
necessary for the ball to clear the tree.
Checklist
✔ Did you show all your
calculations?
✔ Did you draw and label
your graph accurately?
✔ Did you answer all the
questions reflectively, using
complete sentences?
✔ Did you explain your
thinking clearly?
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Chapters
Chapters
4–6
Cumulative Review
Multiple Choice
1. What is the factored form of
12x 2y 3 + 6xy - 2y 2?
A.
B.
C.
D.
2(6x 2y 3 + 3xy - y 2)
2y(6x 2y 2 + 3x - y)
- 2y(- 6x 2y 2 - 3x + y)
all of the above
2. What is the factored form of t 2 + 6t - 27?
A. (t + 9)(t + 3)
B. (t + 9)(t - 3)
C. (t - 9)(t - 3)
D. (t - 9)(t + 3)
3. What is the factored form of 6a 2 + 13a - 5?
A. (2a + 5)(3a - 1) C. (3a + 5)(2a - 1)
B. (6a + 5)(a - 1)
D. (2a - 5)(3a + 1)
4. What is the factored form of 9c 2 + 30c + 25?
A. (3c + 5)(3c - 5) C. (3c + 5)2
B. (9c + 5)(c + 5)
D. (3c - 5)2
5. What is the factored form of 36m 6 - 49?
A. (6m 3 - 7)(6m 3 + 7)
B. (4m 2 + 7)(9m 4 - 7)
C. (6m 3 + 7)2
D. (6m 3 - 7)2
6. What is the factored form of
2xy - 10x + 3y - 15?
A. (y + 5)(2x - 3)
C. (y + 5)(2x + 3)
B. (y - 5)(2x - 3)
D. (y - 5)(2x + 3)
7. Which equation describes the black graph?
y
6
y x2
4
2
x
-4
0
-2
4
-4
-6
A. y = 0.2x 2
B. y = - 2x 2
NEL
C. y = - 0.2x 2
D. y = 2x 2
8. What is the vertex of y = (x + 1)2 - 4, and
what is the equation of the axis of symmetry?
A. (1, - 4); x = 1
C. (1, 4); x = 1
B. ( - 1, - 4); x = - 1 D. ( - 1, 4); x = - 1
9. The following transformations were applied to
the graph of y = x 2: a reflection in the x-axis,
a vertical stretch by a factor of 2, and
a horizontal shift 3 units left. What is the
equation of the transformed graph?
A. y = 2(x + 3)2
C. y = - 2(x - 3)2
2
B. y = - 2x - 3
D. y = - 2(x + 3)2
10. The vertex of a parabola is at (2, 5), and the
parabola passes through (4, - 1). What is the
equation of the parabola?
A. y = - 2.5(x - 2)2 + 5
B. y = - 2.5(x + 2)2 + 5
C. y = - 1.5(x - 5)2 + 2
D. y = - 1.5(x - 2)2 + 5
11. Liz describes the graph of y = - 2(x - 3)2 + 4
using transformations applied to the graph of
y = x 2. Which transformation is not correct?
A. vertical stretch by a factor of 2
B. reflection in the x-axis
C. horizontal shift 3 units left
D. vertical shift 4 units up
12. The daily revenue, R, of a small clothing
boutique depends on the price, d, at which
each dress is sold. R = - 2(d - 135)2 + 1500
models the daily revenue. The owner of the
boutique has discovered that her maximum
daily revenue will increase by $250 if she
increases the price of each dress by $28. What
will be the new daily revenue model?
A. R = - 2(d - 135)2 + 1528
B. R = - 2(d - 385)2 + 1500
C. R = - 2(d - 28)2 + 250
D. R = - 2(d - 163)2 + 1750
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13. Which equation is the vertex form of the
quadratic relation y = 2x(x - 6) - 5?
A. y = 2(x - 3)2 - 23
B. y = 2(x - 6)2 - 5
C. y = 2(x - 3)2 - 5
D. y = 2(x - 3)2 + 23
14. The quadratic relation h = - 5t 2 + 80t models
the height, h, in metres, that an object projected
upward from the ground will reach in t seconds
following its launch. What is the maximum
height that this object will reach?
A. 80 m
C. 320 m
B. 400 m
D. 100 m
15. A quadratic relation can be expressed in three
ways: standard form, factored form, and vertex
form. In which set(s) of equations is a quadratic
relation correctly written in all three forms?
Set 1: y = - 2x 2 + 8x + 10
y = - 2(x + 1)(x - 5)
y = - 2(x - 2)2 + 18
Set 2: y = 3x 2 + 9x - 84
y = 3(x + 7)(x - 4)
y = 3(x + 1.5)2 - 88.5
A. set 1 only
C. neither set
B. set 2 only
D. both sets
16. Which values of x are the solutions to the
equation x 2 + x - 6 = 0?
A.
B.
C.
D.
x = 2, x = 3
x = - 2, x = - 3
x = - 2, x = 3
x = 2, x = - 3
17. Which expression is not a
perfect-square trinomial?
A. 4x 2 + 4x + 1
B. x 2 - 8x + 16
C. x 2 - 10x - 25
D. 9x 2 - 12x + 4
18. Which value for c will make the expression
x 2 - 8x + c a perfect-square trinomial?
A. + 16
C. + 64
B. - 16
D. + 4
366
Cumulative Review
19. Which equation is equivalent to
y = x 2 + 6x + 7?
A. y = (x + 6)2 - 2 C. y = (x + 3)2 - 2
B. y = (x + 3)2 + 7 D. y = (x + 3)2 - 6
20. What are the approximate solutions to the
equation x 2 + 3x - 2 = 0?
A. x = - 3.56, x = 0.56
B. x = - 5.06, x = - 0.94
C. x = - 0.56, x = 3.56
D. x = 1, x = 2
21. Which quadratic equation has exactly
one real root?
A. –9x 2 - 6x + 1 = 0
B. 9x 2 - 6x - 1 = 0
C. 9x 2 - 6x + 1 = 0
D. 9x 2 + 6x - 1 = 0
22. Which quadratic equation has no real solutions?
A. 2x 2 + 5x - 8 = 0
B. -3x 2 + 2x - 5 = 0
C. -x 2 + 2x + 5 = 0
D. 2x 2 + 8x + 3 = 0
23. Which value of k will make the equation
2x 2 + kx + 1 = 0 have two real solutions?
A. k = 1
C. k = 2
B. k = - 4
D. k = - 2
24. The relation h = - 4.9t 2 + 120t + 3 defines
the height of a rocket, where h is the height in
metres and t is the time in seconds following its
launch. If you want to determine how long the
rocket was in flight, what must you do?
A. Determine the vertex.
B. Substitute t = 0, and solve for h.
C. Complete the square.
D. Determine the zeros of the relation.
25. Which quadratic relations have x = 3 as the axis
of symmetry?
A. y = 4(x - 3)2 + 6
B. y = - 2(x - 5)(x - 1)
C. y = x 2 - 6x + 7
D. all of the above
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Cumulative Review
Investigations
Comparing Companies
Sid and Nancy are marketing managers for competing running shoe
companies. They are comparing their annual profit equations in terms
of the number of pairs of shoes manufactured and sold. Sid’s equation is
P = - 6n 2 + 72n - 192. Nancy’s equation is P = - 8n 2 + 40n - 32.
In both equations, P is the profit, in thousands of dollars, and n is the
number of pairs of shoes manufactured and sold, in thousands.
26. Compare the companies in terms of maximum profit, the number
of pairs of shoes manufactured to reach the maximum profit, and
the break-even points.
Penny Drop
At 553 m tall, the CN Tower in Toronto is one of the world’s tallest
self-supporting structures. Suppose that you were standing on the observation
deck at the top of the CN Tower, 447 m above the ground, and you were
able to drop a penny and watch it fall to the ground. This table shows how
the distance of the penny from the ground would change with time.
Time (s)
Distance (m)
0
447.0
2
427.4
4
368.6
6
270.6
8
133.4
27. a) Create a scatter plot, and draw a curve of good fit.
b) Are the data quadratic? Explain.
c) Without using quadratic regression, determine an equation
for your curve of good fit.
d) Using quadratic regression, determine an equation for the curve of
best fit. Compare this equation with your equation for part c).
Comment on the fit.
e) How high is the penny after it has fallen for 5.5 s?
Elaine owns a toy store. She would like to increase the profit on
sales of Silly the Squirrel, which currently sells for $19.99. She has
collected data, through customer surveys, about how different
changes in the price would affect monthly sales. Using her graph,
Elaine used two different strategies to determine a curve of good fit.
She ended up with equations and at the right.
In both equations, x represents the increase or decrease (for negative
x-values) in selling price, and P represents the monthly profit.
28. a) What selling price produces maximum profit
for each equation?
b) What are the break-even prices for each equation?
c) What selling price would you recommend to Elaine?
Explain why.
NEL
Profit vs. Price for Silly the Squirrel
y
2000
1600
Profit ($)
Determining Selling Price
1200
800
400
x
-10 -5
0
Price
decrease ($)
5
10 15 20 25
Price
increase ($)
P = - 10x 2 + 120x + 1600
P = - 11x 2 + 49.5x + 1598
Chapters 4–6
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Chapter
7
Similar
Triangles and
Trigonometry
GOALS
You will be able to
• Determine and compare properties
of congruent and similar triangles
• Solve problems using similar triangles
• Determine side lengths and angle
measures in right triangles using
the primary trigonometric ratios
• Solve problems using right triangle
models and trigonometry
? The distance from Earth to the
Sun can only be measured
indirectly. What are some ways
that you can measure indirectly?
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Getting Started
WORDS YOU NEED to Know
1. Match each term with the example or diagram that best represents it.
a) ratio
b) proportion
i)
c) congruent triangles
d) Pythagorean theorem
iii)
v)
5 cm
3 cm
c
e) hypotenuse
f ) acute angle
A
L
o
o
b
x
x
B
C
M
N
a
4 cm
ii)
3
or 3:5 or 3 to 5
5
iv)
x
3
=
5
15
F
vi)
E
30°
D
SKILLS AND CONCEPTS You Need
Solving a Proportion
Study
Aid
• For more help and practice,
see Appendix A-14.
To solve a proportion, you need to determine the missing value that will
result in an equivalent ratio.
EXAMPLE
Solve each proportion.
x
7
a)
=
3
2
b)
5
2
=
c
9.5
Solution
7
x
=
3
2
a)
x
7
3a b = 3a b
3
2
21
x =
2
x = 10.5
b)
x is divided by 3, so
multiply both sides by 3.
2
5
=
c
9.5
2
5
ca b = ca
b
c
9.5
2 =
5c
9.5
9.5(2) = 9.5 a
19 = 5c
3.8 = c
370
Getting Started
2 is divided by c, so
multiply both sides by c.
5c
b
9.5
Multiply both sides
by 9.5.
Divide both sides by 5.
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Getting Started
2. Solve each proportion.
x
9
=
4
36
2
b
b)
=
5
20
2
5
=
c
7
2d
12
d)
=
9
4
a)
7.8
a
=
5.2
12.0
5.4
4.5
f)
=
y
2.4
c)
e)
Applying the Pythagorean Theorem
The Pythagorean theorem can be used to calculate an unknown side length
in a right triangle if the other two side lengths are known.
Study Aid
• For more help and practice,
see Appendix A-4.
EXAMPLE
7.8 cm
Determine the length of side m.
Solution
Identify the hypotenuse, c.
c = 7.8
m
Communication
Tip
Substitute the given information into
a2 + b 2 = c 2. Square the numbers.
4.12 + m 2 = 7.82
16.81 + m 2 = 60.84
m 2 = 60.84 - 16.81
m 2 = 44.03
m = 144.03
m ⬟ 6.6
Solve for m2. Take the square root
of both sides.
Round the value of m to one
decimal place.
The length of side m
is about 6.6 cm.
3. Determine each unknown side length.
a)
b)
3.6 m
4.1 cm
17.5 cm
c
When calculating side lengths
and angle measures, round
your answers to the same
number of decimal places as
the given information when
the required degree of
accuracy is not stated.
b
14.0 cm
4.8 m
4. The side length of a sugar cube is 2.0 cm. Calculate each distance
to the nearest tenth.
a) from corner to corner across the top
b) from the top corner to the bottom corner
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PRACTICE
Aid
Study
5/8/09
• For help, see the Review
of Essential Skills and
Knowledge Appendix.
Question
Appendix
6
A-15
7, 8
A-16
5. Express each ratio in simplest terms.
a) 10:14
7.5
b)
15
c) - 8:2
27
d)
36
6. In each diagram, determine the measure of x.
a)
c)
107°
42°
74° x
x
b)
d)
x
x
33°
120°
7. State whether the two figures in each pair appear congruent or
similar. Explain how you know.
a)
b)
8. ^ ABC has two 50° angles. What other piece of information do you
need to construct a triangle that is congruent to ^ ABC ?
9. The scale of the building in the diagram at the left is 1:1100.
Calculate the actual height of the building.
10. Complete each sentence as many ways as you can.
3.7 cm
372
Getting Started
a) You find equal angles when . . .
b) Angles add up to 180° when . . .
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Getting Started
APPLYING What You Know
Which Triangles Are the Same?
YOU WILL NEED
Clara has part of a quilt pattern. She wants to know if the different-coloured
triangles are identical.
• grid paper
• protractor
• ruler
• coloured pencils
quilt pattern
enlarged section
?
Which triangles in the pattern are congruent?
A.
Draw the enlarged section. Label the vertices of the three triangles.
B.
Measure the lengths of all the sides in the enlarged section shown above.
Record these lengths on your sketch.
C.
Are any lines in the quilt pattern parallel? Explain how you know, and
indicate these lines on your sketch.
D.
Determine pairs of equal angles in each pair of triangles. Mark the equal
angles on your sketch.
i) yellow triangle and blue triangle
ii) yellow triangle and pink triangle
iii) blue triangle and pink triangle
E.
Which triangles are congruent? Explain how you know.
NEL
Chapter 7
373
7.1
Congruence and Similarity
in Triangles
YOU WILL NEED
GOAL
• dynamic geometry software,
Investigate the relationships between corresponding sides and
angles in pairs of congruent and similar triangles.
or ruler and protractor
INVESTIGATE the Math
B
D
E
A
F
C
Tech
Support
For help constructing triangles,
determining midpoints,
measuring angles and sides,
and calculating using dynamic
geometry software, see
Appendix B-25, B-30, B-26,
B-29, and B-28.
similar triangles
triangles in which corresponding
sides are proportional; similar
triangles are enlargements or
reductions of each other
Colin is a graphic artist. He is creating a logo for a client. He knows that
four new triangles are created when the midpoints of the sides in a triangle
are joined.
?
What are the relationships among the four new triangles
in Colin’s design?
A.
Construct a triangle, and mark the midpoint of each side.
B.
Join the midpoints with line segments and label all the vertices,
as shown in Colin’s design.
C.
Measure all the angles in each of the four small triangles. Measure all the
sides by determining the distances between vertices. What do you notice?
D.
Are the four small triangles congruent? Explain how you know.
E.
Measure all the angles and sides in the large triangle. Compare these
measurements with those for the small triangles. What do you notice?
F.
Are there similar triangles in Colin’s design? If so, identify them.
Explain how you know they are similar.
G.
Explain why it makes sense that the scale factor that relates each small
1
triangle to the large triangle is .
2
H.
Drag one of the vertices in the large triangle.
i) Are the four small triangles still congruent?
ii) Are the small and large triangles still similar? If they are, does
the scale factor change?
I.
Repeat part H several times by choosing different vertices to drag.
scale factor
the value of the ratio of
corresponding side lengths
in a pair of similar figures
Reflecting
J.
374
Are all congruent triangles similar? Are all similar triangles congruent?
Explain.
7.1 Congruence and Similarity in Triangles
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7.1
K.
Suppose that you started with 䉭XYZ and used a scale factor of 2 to
create a similar triangle, 䉭X ¿Y ¿Z ¿ . What would the scale factor be if
you started with 䉭X ¿Y ¿Z ¿ and created 䉭XYZ ? Explain your thinking.
L.
Suppose that you measured two pairs of corresponding angles in two
triangles and discovered that they were equal. Which of these
conclusions could you make? Explain.
i) All the corresponding angles are equal.
ii) The triangles are similar.
iii) The triangles are congruent.
APPLY the Math
EXAMPLE 1
Reasoning about congruence and similarity
a) Create a triangle that is similar but
not congruent to 䉭ABC.
b) Is 䉭DEF similar to 䉭ABC ?
A
3.3 cm
B
2.4 cm
1.2 cm
60°
Gary’s Solution
C
D
60°
3.7 cm
F
1.5 cm
E
I had to create either a larger triangle or a smaller
triangle with the same angles.
a) A ¿C ¿ S 1.2 2.4 2.88 cm
A ¿ B ¿ S 1.2 3.3 3.96 cm
B ¿ C ¿ S 1.2 3.7 4.44 cm
I named the vertices in my new triangle A’, B’, and
C’. I multiplied each side length in 䉭ABC by 1.2
to determine the corresponding side lengths
in the new larger triangle.
A⬘
3.96 cm
B⬘
2.88 cm
60°
4.44 cm
Since C’ corresponds to C, ∠ A’C’B’ must
measure 60°. I constructed A’C’ and measured
a 60° angle. Then I constructed B’C’ and joined
A’ to B’. I measured A’B’ to check that it matched
my calculated value.
C⬘
b)
NEL
2.4
AC
=
= 2
DE
1.2
CB
3.7
=
⬟ 2.47
EF
1.5
∠ C and ∠ E are corresponding angles in the two
triangles. If the triangles are similar, then their
corresponding sides are proportional.
The sides are not proportional, so the triangles
are not similar. ∠ F looks a little greater than
∠ B, and ∠ D looks a little less than ∠ A.
If the angles are different, it makes sense that one
triangle cannot be an enlargement of the other.
Chapter 7
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Connecting similar triangles with the scale factor
Calculate the scale factor that relates the side lengths
of the large logo to the side lengths of the small one.
Monica’s Solution
E
C
30°
8.0 cm
8.0 cm
76°
74°
74°
B
76°
A
C⬘
2.0 cm
I named the vertices in each triangle. Then
I measured all the angles and the long sides
in the triangles in both logos.
30°
D
E⬘
2.0 cm
A⬘ B⬘ D⬘
∠A’ ∠D’ 76˚
∠C’ ∠E’ 30˚
∠A’B’C’ ∠D’B’E’ 74˚
Corresponding angles in the yellow triangles are
equal. Corresponding angles in the green triangles
are equal. Therefore, the yellow triangles are
similar, and so are the green triangles.
䉭ABC ' 䉭A¿B ¿C ¿ and 䉭DBE ' 䉭D ¿B ¿E¿
Showing that there are two pairs of equal angles is
enough to conclude that the triangles are similar,
since the third pair of angles must also be equal.
Since there are two pairs of equal angles, the
triangles are similar.
As well as equal pairs of corresponding angles, the
yellow and green triangles contain a pair of equal
corresponding sides. The large yellow and green
triangles are congruent. The small yellow and green
triangles are also congruent.
In similar triangles, when two corresponding
sides are equal, the ratio of all the corresponding
sides is 1. When the scale factor is 1, the
triangles are congruent.
䉭ABC ⬵ 䉭DBE and 䉭A¿B ¿C ¿ ⬵ 䉭D ¿B ¿E¿
A¿C¿
AC
2.0 1
or
=
8.0 4
Scale factor =
I calculated the ratio of the corresponding sides
in the similar triangles. This calculation is the
same for both the yellow triangles and the
green triangles.
The sides of the large logo have been multiplied by
1
a scale factor of to reduce it to the small one.
4
376
7.1 Congruence and Similarity in Triangles
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7.1
Tip
'
The symbol
is used to indicate similarity. The symbol ⬵ is used to indicate
Communication
congruence. When naming triangles that are congruent or similar, the corresponding
vertices must be listed in the same order. For example, if ∠ A ∠ D, ∠ B ∠ E,
and ∠ C ∠ F, then 䉭ABC ' 䉭DEF.
EXAMPLE 3
Reasoning about similar triangles
to determine side lengths
Show that the two triangles in this
diagram are similar. Then determine
the values of x and y.
D
6.0 cm
C
6.0 cm
8.0 cm
B
3.6 cm
E
x
y
A
Jake’s Solution
∠ BAD ∠ CED
∠ ABD ∠ ECD
So, 䉭ABD ' 䉭ECD.
AB
BD
=
CD
EC
x
14.0
=
6.0
6.0
x
14.0
b = 6.0 a
b
6.0 a
6.0
6.0
14.0 = x
BD
AD
=
ED
CD
y + 3.6
14.0
=
3.6
6.0
3.6 a
Since AB and EC are parallel,
the corresponding angles in
the small and large triangles
are equal.
To calculate x, I set up a
proportion using the
corresponding sides I knew and
the side I needed to determine in
the similar triangles. Then
I solved for x.
To calculate y, I set up another
proportion using the
corresponding sides I knew and
a side that contains y in the similar
triangles. Then I solved for y.
y + 3.6
7.0
b = 3.6 a
b
3.6
3.0
y 3.6 8.4
y 4.8
The value of x is 14.0 cm, and the value of y is 4.8 cm.
NEL
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In Summary
Key Ideas
• If two triangles are congruent, then they are also similar. If two triangles
are similar, however, they are not always congruent.
• If two pairs of corresponding angles in two triangles are equal, then the
triangles are similar. In addition, if two corresponding sides are equal,
then the triangles are congruent.
Need to Know
• If ∠ A ∠ X, ∠ B ∠ Y, and ∠ C ∠ Z, then
AB
BC
AC
=
=
䉭ABC ' 䉭XYZ and
.
XY
YZ
XZ
• If ∠ A ∠ X, ∠ B ∠ Y, and ∠ C ∠ Z, and if AB XY or
BC YZ or AC XZ, then 䉭ABC ⬵ 䉭XYZ.
• When comparing similar triangles, if the scale factor is
• greater than 1, the larger triangle is an enlargement
of the smaller triangle
• between 0 and 1, the smaller triangle is a reduction
of the larger triangle
• 1, the triangles are congruent
X
A
×
¤
B
×
¤
Y
C
Z
X
A
¤
×
B
¤
C
×
Y
Z
CHECK Your Understanding
1. Is 䉭ABC ' 䉭DEF? Justify your answer.
A
B
5.4 cm
36°
67°
5.1 cm
3.3 cm
D 1.8 cm E
36°
1.7 cm 77° 1.1 cm
F
C
2. a) Which triangle is congruent to 䉭ABC ?
b) Which triangles are similar to, but not congruent to, 䉭ABC ?
D
A
4 cm
F
J
¤
3 cm
M
¤
¤
E
8 cm
B
378
10 cm
10 cm
G
2.0 cm
6 cm
C
1.5 cm
H
7.1 Congruence and Similarity in Triangles
I
2.5 cm
K
L
N
6 cm
O
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7.1
3. a) Use grid paper to draw the triangle at the right. Then enlarge it
by a factor of 3.
1
b) Reduce the original triangle by a factor of .
2
PRACTISING
4. i) For each pair of right triangles, determine whether the triangles
K
are congruent, similar, or neither.
ii) If the triangles are congruent, identify the corresponding angles
and sides that are equal. If the triangles are similar, identify
the corresponding angles that are equal, and calculate the scale
factor that relates the smaller triangle as a reduction of the larger
triangle.
a)
c)
D
A
J
G
10 cm
4 cm
H
8 cm
E
B
b)
4 cm
2 cm
I
8 cm
F
6 cm
K
8 cm
L
C
d)
M
R
U
5.0 cm
7.0 m
P
O
N
2.5 cm
4.2 m
S
T
V
Q
5. Are these two triangles similar? Explain how you know.
A⬘
A
11.36 cm
7.26 cm
3.63 cm
C
C⬘
5.68 cm
7.31 cm
B
14.62 cm
B⬘
6. Suppose that 䉭PQR ' 䉭LMN and ∠ P 90°.
a) What angle in 䉭LMN equals 90°? How do you know?
b) If MN 13 cm, LN 12 cm, LM 5 cm, and PQ 15 cm,
what are the lengths of PR and QR ?
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7. Determine the value of each lower-case letter. If you cannot determine
a value, explain why.
a)
c)
A
8m
G
J
z
140°
y
D
H
x
I
30°
B
E
C
b)
a
12.0 cm
¤
11.0 cm
K
F
d) Q
M
c
N
d
c
L
4.0 cm
b
6.0 cm ¤
P
T
S
8.0 cm
R
3.5 cm
e
O
U
8. Determine the value of each lower-case letter.
a)
c)
D
A
h
b
40°
A
E
C
a
80°
5
B
9 cm
cm
g
B
70°
f
4 cm
C
b)
d)
D
D
5 cm
5.0 cm
C
E
A
c d
6.0 cm
A
30°
e
E
C
B
2.0 cm D
i
40°
B
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7.1 Congruence and Similarity in Triangles
E
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7.1
9. Draw three different triangles. Only two of the triangles must be similar.
10. Which type(s) of triangles will always be similar: right, isosceles, or
equilateral?
11. Determine the length of DB.
T
E
C
13 c
2.5 cm
D
m
B
3 cm
A
12. An environmental club is designing a logo using triangles, as shown
A
at the right. If the top and bottom lines of the logo at the right are
parallel, determine the perimeter of the logo.
1.2 cm
1.5 cm
1.5 cm
13. A tree that is 3 m tall casts a shadow that is 2 m long. At the same
time, a nearby building casts a shadow that is 25 m long. How tall
is the building?
14. If two isosceles triangles have one pair of equal angles, are they similar?
C
Explain.
2.6 cm
15. Create a flow chart to summarize your thinking process when you are
determining whether two triangles are congruent or similar.
Extending
16. Follow these steps to prove the Pythagorean theorem using properties
B
of similar triangles:
• First show that the two smaller triangles at the right are similar
to the larger triangle.
• Then use the ratios of corresponding sides to prove the theorem.
17. A right triangle can be partitioned into five smaller congruent
triangles, which are all similar to the original right triangle, as shown
at the right. Determine how this can be done.
C
A
D
1x
18. This trapezoid is an example of self-similarity: the trapezoid is made
of four smaller similar trapezoids. Create your own example of
self-similarity, using a different quadrilateral.
2x
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7.2
YOU WILL NEED
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Solving Similar Triangle
Problems
GOAL
• ruler
Solve problems using similar triangle models.
LEARN ABOUT the Math
B
A new bridge is going to be built across a river, but the width of the river
cannot be measured directly. Surveyors set up posts at points A, B , C , D ,
and E. Then they took measurements relative to the posts.
river
?
9m
A
50 m
C
D
13 m
E
What is the width of the river?
EXAMPLE 1
Selecting a similar triangle strategy
to solve a problem
Use the surveyors’ measurements to determine the width of the river.
Marnie’s Solution
∠ BAC and ∠EDC both equal 90°.
∠ ACB = ∠DCE
So, ^ ABC ' ^DEC .
AB
AC
=
DE
DC
AB
50
=
13
9
50
AB
13 a b = 13a b
13
9
AB ⬟ 72.2
^ ABC and ^ DEC are right
triangles. ∠ACB and ∠DCE are
opposite angles. Since two pairs
of corresponding angles in the
triangles are equal, the triangles
are similar.
I set up a proportion
to determine AB, the width
of the river.
I solved for AB.
The width of the river is about 72 m.
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7.2 Solving Similar Triangle Problems
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7.2
Reflecting
A.
Why do you think the surveyors set up the posts the way they did?
B.
Why do you think the surveyors measured two sides in ^DEC but
only one side in ^ ABC ?
C.
What are the benefits of using similar triangles in this situation?
APPLY the Math
EXAMPLE 2
Solving a problem using a scale diagram
Andrea is a landscape designer. She is working on a backyard that is in the
shape of a right triangle. She needs to cover the yard with sod and then
fence the yard. She starts by drawing a scale diagram using the scale
1 cm represents 6.25 m. She marks the dimensions of the yard on her
drawing as 5 cm, 12 cm, and 13 cm. A roll of sod covers about 0.93 m2. How
many rolls of sod does Andrea need? What length of fencing does she need?
Jordan’s Solution
Let the lengths of the yard be
a , b , and c .
A
b ⫽ 13 cm
C
a ⫽ 5 cm
c ⫽ 12 cm
B
I drew a scale diagram like
Andrea’s using the given
information.
I named each side using the
same letter as the opposite
vertex, but in lower-case. Since
I know that the yard is a right
triangle, the longest side must
be opposite the 90° angle.
Communication
The vertices of a triangle
are usually labelled with
upper-case letters. The
side that is opposite each
angle is labelled with the
corresponding lower-case
letter.
A
c
1 cm
1 cm
S
6.25 m
625 cm
a = (5)(625) c = (12)(625)
a = 3125
c = 7500
Therefore, a is 3125 cm and
c is 7500 cm.
NEL
I converted 6.25 m in the scale to
centimetres, using 100 cm = 1 m.
This allowed me to calculate all
the dimensions in centimetres.
Tip
b
B
a
C
To calculate the area of the yard, I
needed the length of the base
(side a) and the height (side c). I
used my scale factor to calculate
these dimensions of the actual
yard.
Chapter 7
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1
bh
2
1
A = a b(ac)
2
I calculated the area of the yard.
A =
1
= (3125)(7500)
2
= 11 718 750
= 11 718 750 , (100 * 100)
= 1171.875
The area is about 1172 m2.
Number of rolls = 1172 , 0.93
⬟ 1260.2
Andrea needs 1261 rolls of sod.
Since the area covered by a roll
of sod is given in square metres,
I converted the area of the yard
into square metres. To do this, I
divided by the area of a 100 cm
by 100 cm square, which is 1 m2.
I divided the area of the yard by
0.93 to determine the number
of rolls that Andrea needs. I
rounded up since you can't buy
part of a roll.
b = (13)(625)
= 8125
The length of side b is 8125 cm.
I used the scale factor to
calculate the length of side b.
P = a + b + c
= 3125 + 8125 + 7500
= 18 750
= 18 750 , 100
= 187.5
The perimeter is the sum
of a, b, and c.
I divided my answer by 100 to
convert the perimeter to metres.
Andrea needs 187.5 m of fencing.
EXAMPLE 3
Connecting similar triangles to objects and
their shadows
Shiva is standing beside a lighthouse on a sunny day, as shown. She
measures the length of her shadow and the length of the shadow cast
by the lighthouse. Shiva is 1.6 m tall. How tall is the lighthouse?
lighthouse
h
Shiva
1.6 m
4.8 m
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7.2 Solving Similar Triangle Problems
75.0 m
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7.2
Ken’s Solution
␪
First, I had to show that the
triangles are similar.
1.6 m
4.8 m
h
␪
75.0 m
h is the height of the lighthouse.
The angle of elevation, u,
of the Sun is equal in both
diagrams. I assumed that
both Shiva and the lighthouse
were perpendicular to the
ground, so both triangles are
right triangles.
angle of elevation
(angle of inclination)
the angle between the
horizontal and the line of sight
when looking up at an object
The triangles are similar because two
pairs of corresponding angles are equal.
h
75.0
=
1.6
4.8
75.0
b
h = 1.6 a
4.8
h = 25.0
angle of elevation
I set up a proportion of
corresponding side lengths
in the two triangles. Then I
solved my proportion to
calculate the height of the
lighthouse.
Communication
Tip
The symbols u and a are the
Greek letters theta and alpha.
These symbols are often used
to represent the measure of
an unknown angle.
The lighthouse is 25.0 m tall.
In Summary
Key Idea
• Similar triangles can be used to determine lengths that cannot be
measured directly. This strategy is called indirect measurement.
Need to Know
• The ratios of the corresponding sides in similar triangles can be used to
write a proportion. Unknown side lengths can be determined by solving
the proportion.
AB
• If ^ ABC ' ^ DEF and the scale factor is n =
, then the length of
DE
any side in ^ ABC equals n multiplied by the length of the corresponding
side in ^ DEF.
A
CHECK Your Understanding
8 cm
1. a) Explain why you can conclude that ^ ACB ' ^EDB
in the diagram at the right.
b) Determine the scale factor that relates these triangles.
NEL
D
B
2 cm E
C
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2. Miki is standing in a parking lot on a sunny day. He is 1.8 m
tall and casts a shadow that is 5.4 m long.
a) Draw a scale diagram that can be used to measure the angle
at which the Sun’s rays hit the ground.
b) Determine the length of the shadow cast by a nearby tree
that is 12.2 m tall.
3. a) Identify the two similar triangles in the diagram at the left.
A
Explain how you know that these triangles are similar.
b) Determine the value of x .
D
x
5
B
8
E
9
C
PRACTISING
4. Brian is standing near the CN Tower on a sunny day.
K
a) Brian’s height and his shadow form the perpendicular sides
of a right triangle. What does the hypotenuse represent?
1.8 m
4.0 m
1229.5 m
b) The CN Tower casts a shadow that is 1229.5 m long. Explain why
the triangle representing the height of the CN Tower and its shadow
is similar to the triangle representing Brian’s height and his shadow.
c) Determine the scale factor that relates the two triangles.
d) Use the scale factor to determine the height of the CN Tower.
5. How wide is this bay?
A
A
x
18.0 m D
B
30.0 m
C
15.0 m
E
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7.2 Solving Similar Triangle Problems
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7.2
6. On June 30, 1859, Jean François Gravelet crossed the Niagara Gorge
A
on a tightrope. Since he could not measure the distance across
the gorge directly to determine the length of rope he would need,
he used indirect measurement.
a) Explain why ^ DEC is similar to ^ ABC .
b) What ratio might Gravelet have used to determine the scale factor
of the two triangles?
c) Calculate the distance across the gorge.
B
740 m
C
37 m E
26 m
7. A 3.6 m ladder is leaning against a wall, with its base 2 m from the wall.
C
D
a) Draw a scale diagram to represent this situation.
b) Suppose that a 2.4 m ladder is placed against the wall, parallel
to the longer ladder. Explain why the triangles that are formed
by the ground, the wall, and the two ladders are similar.
c) How far up the wall will each ladder reach?
8. Tyler, who is 1.8 m tall, is walking away from a lamppost that is 5.0 m
tall. When Tyler’s shadow measures 2.3 m, how far is he from the
lamppost?
9. A telephone pole is supported by a guy wire, as shown in the diagram at
the right, which is anchored to the ground 3.00 m from the base of the
pole. The guy wire makes a 75° angle with the ground and is attached to
the pole 7.46 m from the top. Another guy wire is attached to the top
of the pole. This guy wire also makes an angle of 75° with the ground
5.00 m from the base of the pole. Determine the height of the pole.
7.46 m
10. The salesclerk at TV-Rama says that the area of a 52 in. plasma screen
is four times as large as the area of a 26 in. screen. Television screens
are measured on the diagonal. Is the salesclerk correct? Explain.
52˝
26˝
75°
75°
3.00 m
5.00 m
11. Surveyors need to determine the height of a hill. They set
T
1.0 m
h
1.6 m
}
NEL
up a laser measuring device on a pole that is 1.0 m tall and
shine the laser toward the top of a second pole, which is
1.6 m tall. Then they adjust the distance between the two
poles until the laser hits the top of the longer pole and the
top of the hill. The 1.6 m pole is 415.0 m from the centre of
the hill. The two poles are 12.0 m apart. Determine the
height of the hill.
415.0 m
12.0 m
Chapter 7
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12. Kent uses a mirror to determine the height of Julie’s window. He
knows that the angle of incidence equals the angle of reflection when
light is reflected off a mirror. How high is the window?
window
vertical line
angle of
incidence
Kent’s
eye
height
1.6 m
angle of reflection
0.5 m
mirror
5.0 m
13. Two lengths in two similar triangles are given.
A
5.2 cm
3.4 cm
B
y
C F
D
x
E
a) Which lengths do you need to know, other than x and y ,
to determine x and y ?
b) Explain how you would use these other lengths to determine x and y .
Extending
14. Determine the width of this river, if AB = 96 m, AC = 204 m, and
BD = 396 m.
A
C
D
B
E
15. A square photo, with an area of 324.00 cm2, is in a square frame
that has an area of 142.56 cm2. The dimensions of the photo and
the frame are proportional.
a) Determine the scale factor that relates the dimensions of the photo
and the frame.
b) Determine the width of the frame.
388
7.2 Solving Similar Triangle Problems
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
How do you know whether two triangles are similar,
congruent, or neither?
A:
Since congruent triangles are the same size and shape, all the
corresponding side lengths and angle measures are equal. Since similar
triangles are the same shape but different sizes, the corresponding
angles are equal and the corresponding side lengths are proportional.
If the corresponding angles of two triangles are not equal, then the
triangles are neither similar nor congruent.
A
Q
^ ADE ' ^ABC
x
x
^ ADE ' ^QRS
o
D
E
^ ABC ⬵ ^QRS
o
B
Study
Aid
• See Lesson 7.1, Examples 1
and 2.
• Try Mid-Chapter Review
Question 1.
o
C
R
Q:
How can the properties of similar triangles be used
to calculate an unknown side length in a triangle?
A:
After you determine that two triangles are similar, you can set up
a proportion using corresponding sides. Solving the proportion
gives the unknown side length.
S
Study
Aid
• See Lesson 7.1, Example 3,
and Lesson 7.2, Examples 1
to 3.
• Try Mid-Chapter Review
Questions 3, and 5 to 10.
EXAMPLE
^ABC ' ^DEF . Determine the value of x.
Solution
AC
AB
=
DE
DF
5.0
4.0
=
x
3.0
AB corresponds to DE, and AC
corresponds to DF. Since the
triangles are similar, the ratios
of these sides are equal.
A
4.0 cm
¤
␪
5.0 cm
C
D
x
3.0 cm
¤
␪ F
o
o
E
Solve for x.
B
xa
5.0
4.0
b
b = xa
x
3.0
4.0x
5.0 =
3.0
(3.0)(5.0) = 4.0x
15.0
= x
4.0
3.75 = x
x is about 3.8 cm.
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Chapter 7
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PRACTICE Questions
Lesson 7.1
5. Determine the values of x and y.
1. Name the triangles that are
a) congruent to ^ ABC
b) similar to ^ ABC
¤
¤
x
4.1
o
7.0
5.3
o
C
y
B
E
6. For ^ ABC ' ^ DEF :
A
O
N
F
6.8
M
I
a) Determine the length of BC.
b) Determine the length of DE.
c) Is ^ GHI ' ^ DEF ? Explain.
A
G
D
H
4
G
D
C
a) ∠ ABC = .
d) ^STR ' .
b) ∠ BCA = .
e)
AB
= .
RS
ST
= .
BC
f ) ∠ SRT = .
3. Write a proportion for the corresponding side
lengths in these similar triangles.
x
a
c
o
f
x
b
d
B
12
2. ^ ABC ' ^ RST. Complete each statement.
c)
6
F
8
15
E
I
6
H
Lesson 7.2
7. Nora, who is 172.0 cm tall, is standing near a
tree. Nora’s shadow is 3.2 m long. At the same
time, the shadow of the tree is 27.0 m long.
How tall is the tree?
8. A right triangle has side lengths 6 cm, 8 cm, and
10 cm. The longest side of a larger similar
triangle measures 15 cm. Determine the
perimeter and area of the larger triangle.
9. Connie placed a mirror on the ground, 5.00 m
e
o
4. Peter says, “If you know the measures of two
angles in each of two triangles, you can always
determine if the triangles are similar.” Is this
statement true or false? Explain your reasoning.
390
Mid-Chapter Review
from the base of a flagpole. She stepped back
until she could see the top of the flagpole reflected
in the mirror. Connie’s eyes are 1.50 m above the
ground and she saw the reflection when she was
1.25 m from the mirror. How tall is the flagpole?
10. Cam is designing a new flag for his hockey team.
The flag will be triangular, with sides that
measure 0.8 m, 1.2 m, and 1.0 m. Cam has
created a scale diagram, with sides that measure
20 cm, 30 cm, and 25 cm, to take to a flag maker.
Did Cam create his scale diagram correctly?
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Exploring Similar
Right Triangles
YOU WILL NEED
GOAL
Explore the connection between the ratios of the sides in the same
triangle for similar triangles.
• dynamic geometry software,
or ruler and protractor
EXPLORE the Math
Mark noticed a common design element in the wheelchair ramp that was
installed at his home and the skateboard ramp at a park. The ground, the
ramp, and the vertical supports formed a series of right triangles.
?
How are the ratios of the sides in similar right triangles related?
A.
Using dynamic geometry
software, or a protractor and
a ruler, construct a diagram
like the one shown. Make sure
that ∠ A = 40°. Also make
sure that all the vertical lines
are perpendicular to AB .
C
I
the side that is directly across from
a specific acute angle in a right
triangle; for example, BC is the
opposite side in relation to ∠A
C
G
hypotenuse
side
opposite
to A
A
B
E
40°
A
D
F
H
B
B.
Explain why the four right triangles in your diagram are similar.
C.
For each triangle in your diagram, measure the lengths of the opposite
side and adjacent side, as well as the hypotenuse. Record these values
in a table like the one at the top of the next page. Calculate each ratio
to two decimal places.
NEL
opposite side
side adjacent to A
adjacent side
the side that is part of an acute
angle in a right triangle, but is
not the hypotenuse; for
example, AB is the adjacent side
in relation to ∠A above.
Chapter 7
391
Trigonometric Ratios
Triangle
Side
Opposite
to ∠A
Side
Adjacent
to ∠A
Hypotenuse
^ ABC
BC = .
AB = .
AC = .
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
BC
= .
AC
AB
= .
AC
BC
= .
AB
^ ADE
^ AFG
^ AHI
Tech
Support
For help constructing triangles,
measuring angles and sides,
and calculating using dynamic
geometry software, see
Appendix B-25, B-26, B-29,
and B-28.
D.
Describe the relationships in your table.
E.
Do you think the relationships you described for part D would change
if ∠ A were changed to a different measure? Make a conjecture, and then
test your conjecture by creating a new diagram using a different acute
angle for ∠ A . Use the measure of this angle to repeat part C.
F.
Compare your results with your classmates’ results. Summarize what
you discovered.
Reflecting
G.
In each triangle, the ratio
opposite
is equivalent to the slope of AC.
adjacent
Explain why.
H.
Did the relationships involving the ratios of the sides in similar right
triangles depend on the size of ∠ A ? Explain.
In Summary
Key Idea
• In similar right triangles, the following ratios are equivalent
for the corresponding acute angles:
A
opposite
AB
DE
S
=
adjacent
BC
EF
hypotenuse
opposite
AB
DE
opposite
S
=
hypotenuse
AC
DF
adjacent
BC
EF
S
=
adjacent
C
B
hypotenuse
AC
DF
D
hypotenuse
opposite
F adjacent E
Need to Know
• The slope of a line segment is related to the angle that the line segment
makes with the x-axis.
• If two lines make the same angle with the x-axis, they have the same slope.
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7.3 Exploring Similar Right Triangles
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7.3
FURTHER Your Understanding
1. a) For each triangle, state the opposite side and adjacent side to u
and the hypotenuse.
F
C
A
20 cm
12 cm
5 cm
␪
4 cm
3 cm
B
␪
D
16 cm
E
b) For ^ ABC ' ^ DEF , show that these ratios are equal for u
in both triangles.
i) opposite : hypotenuse
ii) adjacent : hypotenuse
iii) opposite : adjacent
2. a) Identify the triangles that are similar to ^ ABC .
A
K
I
G
E
B
J
H
F
D
C
b) State each ratio for all the triangles. Use ∠C when identifying
the opposite and adjacent sides.
i) opposite : hypotenuse
ii) adjacent : hypotenuse
iii) opposite : adjacent
c) State all the ratios for part b) that are equal.
3. a) Part of a road rises 8 m over a run of 120 m. What is
the rise over a run of 50 m if the slope remains constant?
b) Compare the slopes for part a). Explain why these slopes
are the same.
4. A moving truck has two ramps, 3 m long and 4 m long,
for loading and unloading. Which ramp creates a greater
angle of elevation with the ground? Include a diagram
in your answer.
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The Primary Trigonometric
Ratios
GOAL
Determine the values of the sine, cosine, and tangent ratios
for a specific acute angle in a right triangle.
LEARN ABOUT the Math
Nadia wants to know the slope of a ski hill. Her trail map shows that the
hill makes an angle of 18° with the horizontal. Her friend suggests that she
use trigonometry to calculate the slope.
?
How can the 18° angle be used to determine the slope
of the ski hill?
EXAMPLE 1
Connecting an angle to the ratios of the
sides in a right triangle
trigonometry
the branch of mathematics that
deals with the properties of
triangles and calculations based
on these properties
Communication
Tip
Opposite and adjacent sides
are named relative to a specific
acute angle in a triangle.
Use the angle of the ski hill to determine the slope of the ski hill.
Nadia's Solution
18°
run
hypotenuse
18°
adjacent
rise
opposite
tangent
the ratio of the length of the
opposite side to the length of
the adjacent side for either acute
angle in a right triangle; the
abbreviation for tangent is tan
Tech
Support
When using a scientific
calculator to calculate the
tangent ratio, first make sure
that you are in degree mode.
394
opposite
adjacent
rise
tan 18° =
run
tan A =
tan 18° ⬟ 0.3249
The slope of the ski hill
is approximately 0.32.
7.4 The Primary Trigonometric Ratios
I drew a diagram to represent
this situation.
rise
ratio is the
run
same in any right triangle with
an 18° angle. So I divided the
length of the side opposite to
the 18° angle by the length of
the side adjacent to it.
I knew that the
I knew that I could describe the
opposite
ratio
using tangent.
adjacent
I used my calculator to
determine tan 18°.
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7.4
Reflecting
A.
How did Nadia use the properties of similar triangles when she used
the tangent ratio to solve the problem?
B.
Why did Nadia use the tangent of the 18° angle instead of the
72° angle to solve the problem?
APPLY the Math
EXAMPLE 2
Connecting each trigonometric ratio
to an acute angle
A
Determine the values of the tangent, sine, and
cosine ratios for ∠ A and ∠ B to four decimal places.
67°
23°
B
Paula’s Solution
c
67° b
a
C
23°
A
c ⫽ hypotenuse
a ⫽ opposite
B
67°
b ⫽ adjacent
First, I labelled each side
using the lower-case
letter that matched the
angle opposite the side.
the ratio of the length of the
adjacent side to the length of
the hypotenuse for either acute
angle in a right triangle; the
abbreviation for cosine is cos
I named the sides
relative to ∠A as
opposite, adjacent, and
hypotenuse.
primary trigonometric ratios
C
opposite
a
=
c
hypotenuse
sin 67° ⬟ 0.9205
adjacent
b
=
cos A =
c
hypotenuse
cos 67° ⬟ 0.3907
opposite
a
=
tan A =
adjacent
b
tan 67° ⬟ 2.3559
sin A =
To four decimal places, sin A is 0.9205,
cos A is 0.3907, and tan A is 2.3559.
NEL
the ratio of the length of the
opposite side to the length of
the hypotenuse for either acute
angle in a right triangle; the
abbreviation for sine is sin
cosine
A
B
C
sine
I determined the three
primary trigonometric
ratios for ∠ A.
the basic ratios of trigonometry
(sine, cosine, and tangent)
Tech
Support
When using a scientific
calculator to calculate the
primary trigonometric ratios,
press SIN , COS , or
TAN and then enter the
angle, or enter the angle and
then press SIN , COS , or
TAN , depending on your
calculator.
Chapter 7
395
A
c ⫽ hypotenuse
b ⫽ opposite
C
Communication
Tip
When necessary, trigonometric
ratios are usually expressed
with four decimal places of
accuracy. This is done to
calculate the angles with
enough precision. Carrying as
many digits as possible until
the final step in a calculation
reduces the possibility of
variations in answers due to
rounding.
23°
a ⫽ adjacent
I renamed the sides relative to
∠B as opposite, adjacent, and
hypotenuse.
B
opposite
b
=
c
hypotenuse
sin 23° ⬟ 0.3907
adjacent
a
=
cos B =
c
hypotenuse
cos 23° ⬟ 0.9205
opposite
b
=
tan B =
a
adjacent
tan 23° ⬟ 0.4245
sin B =
I determined the three primary
trigonometric ratios for ∠B.
To four decimal places, sin B is 0.3907,
cos B is 0.9205, and tan B is 0.4245.
EXAMPLE 3
Connecting an acute angle to the sides
in a right triangle
Determine the measure of u to the nearest degree, using each primary
trigonometric ratio.
2.00 cm
Jim’s Solution
opposite
hypotenuse
2.00
sin u =
5.39
2.00
b
u = sin - 1 a
5.39
u ⬟ 21.8°
u ⬟ 22°
inverse
the reverse of an original
statement; for example, if
x = sin u, the inverse is
u = sin - 1x
Tech
Support
When using a scientific
calculator to calculate
the inverse sine, press
2ND
SIN
and enter
the ratio, or enter the ratio
and press
396
2ND
SIN .
5.39 cm
x
␪
sin u =
I started by determining u using
the sine ratio. If I knew the
2.00
angle, its sine would be
.
5.39
Since I knew the sine and not
the angle, I had to use the
inverse of sine, which is sin - 1.
x 2 + 2.002 = 5.392
x 2 + 4.00 = 29.0521
x 2 = 29.0521 - 4.00
x 2 = 25.0521
x = 225.0521
x ⬟ 5.01
Since the tangent and cosine
ratios both involve the adjacent
side, x, I calculated the length
of this side using the
Pythagorean theorem.
7.4 The Primary Trigonometric Ratios
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7.4
adjacent
hypotenuse
5.01
cos u =
5.39
5.01
b
u = cos - 1 a
5.39
u ⬟ 21.6°
u ⬟ 22°
Tech
cos u =
opposite
adjacent
2.00
tan u =
5.01
2.00
b
u = tan - 1 a
5.01
u ⬟ 21.8°
u ⬟ 22°
tan u =
To determine the angle using
the cosine ratio, I had to use
the inverse of cosine, cos - 1. My
answer is the same as the one
I calculated using sine.
Support
For help using a TI-83/84
graphing calculator to
calculate trigonometric ratios
and determine angles, see
Appendix B-12. If you are
using a TI-nspire, see
Appendix B-48.
To determine the angle using
the tangent ratio, I had to use
the inverse of tangent, tan - 1.
All the primary trigonometric
ratios gave me the same
answer. Next time, I know that
I only have to use one of them
to determine u.
In Summary
Key Ideas
• The primary trigonometric ratios for ∠A are sin A, cos A, and tan A.
• If ∠A is one of the acute angles in a right triangle, the primary
trigonometric ratios can be determined using the ratios of the sides.
Need to Know
• If ∠A is one of the acute angles in a right triangle, the three primary
trigonometric ratios for ∠A can be written as
B
opposite
sin A =
hypotenuse
hypotenuse
adjacent
opposite
cos A =
hypotenuse
opposite
tan A =
A adjacent C
adjacent
2
• Using the Pythagorean theorem, opposite + adjacent2 = hypotenuse2
in any right triangle.
NEL
Chapter 7
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CHECK Your Understanding
A
1. a) Which side is opposite to ∠ A?
b) Which side is adjacent to ∠A?
c) Which side is the hypotenuse?
C
2. Determine the value of each ratio
to four decimal places.
a) tan 34°
b) sin 78°
B
c) cos 49°
d) sin 12°
3. Determine the measure of u to the nearest degree.
a) sin u = 0.5
b) tan u = 1
c) cos u = 0.5
d) sin u = 0.8660
PRACTISING
4. ^XYZ is a right triangle, with ∠ X = 90°.
a) Sketch ^XYZ. Label the sides using lower-case letters.
b) Write the ratios for sin Y, cos Y, and tan Y in terms of x, y, and z.
5. Determine each ratio, and write it as a decimal to four decimal places.
A
a) sin C
d) tan C
b) cos C
c) tan B
e) cos B
f ) sin B
6. Decide whether each statement is true
or false. Justify your decision.
a) sin u = 0.4
b) tan a = 2
c) cos a ⬟ 0.8929
d) cos u ⬟ 0.8929
15 cm
8 cm
17 cm
C
␣
2.5 m
B
5.6 m
␪
5.0 m
7. a) Calculate the measure of x in the diagram at the left to the nearest
x
65
25
degree, using one of the primary trigonometric ratios.
b) Do you need to use a primary trigonometric ratio to determine
the measure of y? Explain.
y
60
8. Solve for x, and express your answer to one decimal place.
x
6
x
b) sin 62° =
14
a) cos 45° =
␪
398
15
x
10
f ) sin 45° =
x
e) cos 60° =
9. Identify the primary trigonometric ratio for u that is equal to each
50
K
54
x
20
12
d) tan 80° =
x
c) tan 75° =
20
ratio for the triangle at the left.
50
20
a)
b)
54
50
7.4 The Primary Trigonometric Ratios
c)
20
54
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7.4
10. Determine the length of x. Then state the primary trigonometric ratios
A
for u.
a)
b)
␪
8 cm
12 cm
x
␪
x
13 cm
15 cm
11. Determine the measure of u, to one decimal place, for each triangle
in question 10.
12. Determine the measure of u, to one decimal place.
2
5
4
b) cos u =
9
a) sin u =
c) tan u = 3
d) sin u =
1
2
1
mean that the side adjacent to the 60° angle
2
C
measures 1 unit and the hypotenuse measures 2 units? Explain.
13. Does cos 60° =
14. Draw two different right triangles for which tan u = 1. Determine
T
the measurements of all the sides and angles. Then compare
the two triangles.
15. a) Could the orange side in ^ ABC at the right be considered an
C
adjacent side when determining a trigonometric ratio? Explain.
b) Could the orange side be considered an opposite side when
determining a trigonometric ratio? Explain.
c) Could the orange side be considered the hypotenuse when
determining a trigonometric ratio? Explain.
A
B
Extending
16. For what value of u does sin u = cos u ? Include a diagram
in your answer.
17. Explain why the value of tan u increases as the measure of u increases.
␪
NEL
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Solving Right Triangles
GOAL
Use primary trigonometric ratios to calculate side lengths and
angle measures in right triangles.
LEARN ABOUT the Math
A farmers’ co-operative wants to buy
and install a grain auger. The auger
would be used to lift grain from the
ground to the top of a silo. The greatest
angle of elevation that is possible for the
auger is 35°. The auger is 18 m long.
?
What is the maximum height
that the auger can reach?
EXAMPLE 1
18 m
h
35°
Solving a problem for a side length
using a trigonometric ratio
Calculate the maximum height that the auger can reach.
Hong’s Solution
hypotenuse
18 m
opposite
h
35°
adjacent
opposite
hypotenuse
h
sin 35° 18
sin u Tech
Support
For help using a TI-83/84
graphing calculator to
calculate trigonometric ratios,
see Appendix B-12. If you are
using a TI-nspire, see
Appendix B-48.
400
18 (sin 35°) 18a
h
b
18
18 (sin 35°) h
10 ⬟ h
I drew a diagram to model the
problem. The height is the length
of the side that is opposite the
35° angle. I named the other sides
relative to the 35° angle.
Because I knew the length of the
hypotenuse, I used the sine ratio.
The sine of 35° equals the opposite
side, or height, divided by the
hypotenuse.
I multiplied both sides by 18 and
evaluated 18 (sin 35°) using a
calculator. I rounded my answer
using the degree of accuracy
in the other measures.
The maximum height that the auger
can reach is about 10 m.
7.5 Solving Right Triangles
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Reflecting
A.
If the height of the grain auger is increased, what happens to the sine,
cosine, and tangent ratios for the angle of elevation? Explain.
B.
Why can you use either the sine ratio or the cosine ratio to calculate
the maximum height?
C.
Explain why Hong might have chosen to use the sine ratio.
APPLY the Math
EXAMPLE 2
Connecting the cosine ratio with the length of the hypotenuse
Determine the length of p.
55 mm
15°
p
Mandy’s Solution
adjacent
55 mm
15°
p
hypotenuse
opposite
adjacent
hypotenuse
55
cos 15° =
p
cos u =
p (cos 15°) = p a
55
b
p
p (cos 15°) = 55
p (cos 15°)
55
=
cos 15°
cos 15°
55
p =
cos 15°
p ⬟ 57
The 55 mm side is adjacent to the 15° angle.
I named the rest of the sides in the triangle
relative to the 15° angle.
Because I knew the adjacent side and had to
determine the hypotenuse, I used the cosine
ratio. The cosine of 15° equals the adjacent side
divided by the hypotenuse.
I multiplied both sides by p. Then I divided both
sides by cos 15° to solve for p. I rounded my
answer to the nearest millimetre.
The length of p is about 57 mm long.
NEL
Chapter 7
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EXAMPLE 3
Connecting the cosine ratio with an angle
measure
Noah is flying a kite and has released 25 m of string. His sister is standing
8 m away, directly below the kite. What is the angle of elevation of the string?
Jacob’s Solution
hypotenuse
25 m
I drew a right triangle showing
Noah, his sister, and the kite. I
labelled the triangle with the
information that I knew and
named the sides of the triangle
relative to the angle of elevation.
opposite
8m
adjacent
Noah
cos u =
sister
Because I knew the lengths of the
adjacent side and the hypotenuse,
I used the cosine ratio.
8
25
u = cos - 1 a
8
b
25
I used the inverse cosine
to determine the angle.
u ⬟ 71°
The angle of elevation of the string
is about 71°.
EXAMPLE 4
Communication
Tip
To solve a triangle means to
determine all unknown angle
measures and side lengths.
Selecting a trigonometric strategy to solve
a triangle
Solve 䉭 ABC, given ∠ A 90°, a 7.8 m, and c 5.2 m.
Chloe’s Solution
B
I drew the triangle and labelled it
with the measurements that I knew.
a 7.8 m
c 5.2 m
A
402
I rounded my answer
to the nearest degree.
7.5 Solving Right Triangles
C
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7.5
c
a
5.2
cos B =
7.8
I started with ∠ B. Since side c is
adjacent to ∠ B and side a is the
hypotenuse, I used the cosine
ratio.
cos B =
∠ B = cos - 1 a
5.2
b
7.8
I rounded my answer to the
nearest degree.
∠ B ⬟ 48°
∠ C ⬟ 180° - 90° - 48°
∠ C ⬟ 42°
b2 + c 2 = a2
b 2 + 5.22 = 7.82
b 2 + 27.04 = 60.84
b 2 = 60.84 - 27.04
b 2 = 33.80
b = 233.80
b ⬟ 5.8
I knew the sum of the angles in a
triangle is 180°. I used this to
determine ∠ C.
I used the Pythagorean theorem
to solve for b. I could have used
the sine or tangent ratios instead.
In ^ ABC, ∠ B ⬟ 48°, ∠ C ⬟ 42°,
and b ⬟ 5.8 m.
In Summary
Key Idea
• Trigonometric ratios can be used to calculate unknown side lengths and
unknown angle measures in a right triangle. The ratio you use depends
on the information given and the quantity you need to calculate.
Need to Know
• To determine the length of a side in a right triangle using trigonometry,
you need to know the length of another side and the measure of one
of the acute angles.
• To determine the measure of one of the acute angles in a right triangle
using trigonometry, you need to know the lengths of two sides.
CHECK Your Understanding
1. Solve for x, to one decimal place, using the indicated trigonometric ratio.
a) cosine
4.3 km
b) tangent
x
54°
25.0 cm
60°
x
NEL
Chapter 7
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2. Determine the value of u, to the nearest degree, in each triangle.
a)
b)
3 cm
8 cm
5 cm
␪
␪
3. Solve 䉭 ABC.
A
5 cm
␪
18 cm
b
37°
C
a
B
PRACTISING
4. Solve for ∠ A to the nearest degree.
a) sin A 0.9063
b) cos A 4
5
5. For each triangle,
K
i) state two trigonometric ratios you could use to determine x
ii) determine x to the nearest unit
a)
b)
50°
39°
90 mm
x
x
40°
51°
30 cm
6. For each pair of side lengths, calculate the measure of u to the nearest
b
c
␪
a
degree for the triangle at the left.
a) a 10 and c 10 b) b 12 and c 6 c) a 9 and b 15
7. Using trigonometry, calculate the measures of ∠ A and ∠ B in each
triangle. Round your answers to the nearest degree.
a) A
b) C
16
20
A
37
32
B
C
404
7.5 Solving Right Triangles
B
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7.5
8. Calculate the measure of the indicated angle, to the nearest degree,
in each triangle.
a) In 䉭ABC, ∠ C 90°, a 11.3 cm, and b 9.2 cm. Calculate ∠ A.
b) In 䉭DEF, ∠ D 90°, d 8.7 cm, and f 5.4 cm. Calculate ∠ F.
9. Janice is getting ready to climb a steep cliff. She needs to fasten herself to
a rope that is anchored at the top of the cliff. To estimate how much rope
she needs, she stands 50 m from the base of the cliff and estimates that
the angle of elevation to the top is 70°. How high is the cliff?
10. Solve for i and j.
T
i
j
70°
50 m
60°
8 cm
11. A ladder leans against a wall, as shown. How long is the ladder,
A
to the nearest tenth of a metre?
2.8 m
75°
12. Kelsey made these notes about 䉭 ABC. Determine whether
C
each answer is correct, and explain any errors.
A
25.5 cm
12.0 cm
28°
B
12.0
25.5
b) ∠ A 62°
a) sin A NEL
22.5 cm
C
25.5
22.5
d) tan A 1.875
c) cos C 24
51
f ) tan C 0.53
e) sin C Chapter 7
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13. Solve each triangle. Round the measure of each angle to the nearest
degree. Round the length of each side to the nearest unit.
a) A
8 mm
b) J
B
c)
72°
Q
10 cm
5 mm
K
L
13 km
C
42°
R
S
14. For a ladder to be stable, the angle that it makes with the ground
should be no more than 78° and no less than 73°.
a) If the base of a ladder that is 8.0 m long is placed 1.5 m from
a wall, will the ladder be stable? Explain.
b) What are the minimum and maximum safe distances from
the base of the ladder to the wall?
15. a) Create a mind map that shows the process of choosing the correct
trigonometric ratio to determine an unknown measure in a right
triangle.
b) Does the process differ depending on whether you are solving
for a side length or an angle measure? Explain.
Extending
16. Determine the diameter of the circle, if O is the centre of the circle.
28 mm
O
17. a) Determine the exact value of x in the
60°
triangle at the right using trigonometry.
1
b) Determine the exact value of y using
the Pythagorean theorem.
c) Determine the sine, cosine, and tangent
ratios of both acute angles. What do
you notice?
y
30°
x
18. a) Draw a right isosceles triangle.
b) Calculate the sine and cosine ratios for one of the acute angles.
Explain your results.
406
7.5 Solving Right Triangles
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Curious Math
Curious Math
A Unit That Did Not Measure Up
Because sailors and pilots are not travelling on land, they use nautical miles
to measure distances. Originally, a nautical mile (M) was the distance
1
across one minute of latitude. One minute ( ¿ ) is of a degree (°).
60
N
90°
latitude
75°
60°
45°
30°
17° 0 (16° 60)
16° 45
0°
16° 30
15°
16° 15
30°
45°
16° 0
minutes of degrees
15°
60°
longitude
S
Scientists later discovered that Earth bulges at the equator and is flatter
at the poles. So, the nautical mile is shorter as you approach the equator
and longer as you approach each pole.
1. a) Write 25°16 ¿ as a decimal.
b) Write 48.30° using degrees and minutes.
2. Determine the length, M, in metres, of the original nautical mile
at each location. Use the formula M 1852.27 9.45 (cos 2u),
where u is the latitude in degrees.
a) Kingston, Ontario: latitude 44°15 ¿
b) Yellowknife, Northwest Territories: latitude 62°28 ¿
c) Alert, Nunavut: latitude 82°30 ¿
3. The nautical mile was internationally redefined in 1929 as being
exactly 1852 m. Explain why this value might have been chosen.
4. What does the expression 9.45 (cos 2u) mean in the formula
M 1852.27 9.45 (cos 2u )?
NEL
Chapter 7
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Solving Right Triangle
Problems
GOAL
Use the primary trigonometric ratios to solve problems
that involve right triangle models.
LEARN ABOUT the Math
Jackie works for an oil company. She needs to drill a well to an oil deposit.
The deposit lies 2300 m below the bottom of a lake, which is 150 m deep.
The well must be drilled at an angle from a site on land. The site is 1000 m
away from a point directly above the deposit.
?
At what angle to Earth’s surface should Jackie drill the well?
EXAMPLE 1
Solving a problem using a right triangle
model
Determine the angle at which the well should be drilled.
Jackie's Solution
site
I drew a diagram that shows
where the lake, oil deposit, and
drill site are located.
1000 m top of lake
␪
150 m
bottom of lake
angle of depression
(angle of declination)
2300 m
the angle between the
horizontal and the line of sight
when looking down at an object
angle of depression
object
d
I added a line to show the angle
of depression to the deposit.
I labelled the angle of
depression u.
oil deposit
d = 150 + 2300
= 2450
2450
1000
tan u ⫽ 2.45
u ⫽ tan-1 (2.45)
u ⬟ 68°
tan u ⫽
I calculated the length of the
side that is opposite angle u by
adding the two given vertical
distances.
To calculate u, I used my
calculator. Since I knew the
opposite and adjacent sides to u,
I used the inverse tangent.
The well should be drilled
at an angle of about 68°.
408
7.6 Solving Right Triangle Problems
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7.6
Reflecting
A.
How does an angle of depression relate to an angle of elevation?
B.
How could Jackie calculate the distance from the oil deposit
to the drill site?
APPLY the Math
EXAMPLE 2
Solving a problem using a clinometer
Ayesha is a forester. She uses a clinometer (a device used to measure
angles of elevation) to sight the top of a tree. She measures an angle of
48°. She is standing 7.2 m from the tree, and her eyes are 1.6 m above
ground. How tall is the tree?
48°
1.6 m
7.2 m
Joan’s Solution
hypotenuse
48°
1.6 m
h opposite
adjacent
7.2 m
1.6 m
opposite
adjacent
h
tan 48° ⫽
7.2
tan u ⫽
7.2 (tan 48°) ⫽ 7.2a
h
b
7.2
7.2 (tan 48°) ⫽ h
8.00 ⬟ h
tree height ⫽ 1.6 ⫹ 8.0
⫽ 9.6
The tree is about 9.6 m tall.
NEL
I drew a diagram to model the problem. The height
is the length of the side that is opposite the 48°
angle. I named the rest of the sides in the triangle
relative to the 48° angle.
Because I knew the adjacent side, I used the tangent
ratio. The tangent of 48° equals the opposite side, or
height, divided by the adjacent side.
I multiplied both sides by 7.2 and evaluated.
I added the distance from the ground to Ayesha’s
eyes to the height of the triangle to calculate the
height of the tree.
Chapter 7
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Solving an area problem when the height is unknown
A group of students are on an outdoor education trip. They leave their
campsite and travel 240 m before reaching the first orienteering
checkpoint. They turn, creating a 42° angle with their previous path,
and travel another 180 m to get to the second checkpoint. They turn
again and travel the shortest possible path back to their campsite.
What area of the woods did their triangular route cover?
Safety Connection
A compass, map, first-aid
kit, and signal device are
important pieces of equipment
when hiking in the woods.
Hugo’s Solution
2nd checkpoint
180 m
42°
1st checkpoint
I created a diagram to represent the situation.
240 m
campsite
2nd checkpoint
180 m
42°
1st checkpoint
I had to determine the height of the triangle to
calculate its area. I drew a line perpendicular to
the 240 m side. I labelled this line as h.
h
240 m
campsite
h
sin 42° ⫽
180
180 (sin 42°) ⫽ h
120.4 ⬟ h
The height of the triangle is about 120.4 m.
1
A ⫽ bh
2
1
A ⫽ (240)(120.4)
2
A ⫽ 14 448
In the right triangle that contains the 42° angle,
h is the side opposite this angle. I knew the
hypotenuse, so I used the sine ratio to solve for h.
I calculated the area.
The area of the woods that the triangular
route covered was about 14 448 m2.
410
7.6 Solving Right Triangle Problems
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7.6
EXAMPLE 4
Solving a problem using two right triangles
Lyle stood on land, 200 m away from one of the towers on a bridge.
He reasoned that he could calculate the height of the tower by
measuring the angle to the top of the tower and the angle to its base
at water level. He measured the angle of elevation to its top as 37°
and the angle of depression to its base as 21°. Calculate the height
of the tower from its base at water level, to the nearest metre.
Jenna’s Solution
a
L
21°
200 m
37°
roadway
b
a
200
200 (tan 37°) ⫽ a
150.7 ⬟ a
b
tan 21° ⫽
200
200 (tan 21°) ⫽ b
76.8 ⬟ b
tan 37° ⫽
height ⫽ a ⫹ b
⫽ 150.7 ⫹ 76.8
⫽ 227.5
I started by drawing a diagram of the bridge and
tower, and labelling the given angles. I split the
height of the tower in two. I named the upper part
of the tower (above roadway) a and the lower part
of the tower (below roadway) b. This created two
right triangles.
I had to determine the side that is opposite the 37°
angle in the top triangle. I knew the adjacent side,
so I used the tangent ratio.
I had to determine the side that is opposite the
21° angle in the bottom triangle. I knew the
adjacent side, so I used the tangent ratio again.
I calculated the height of the tower by adding
a and b.
The tower is about 228 m tall from its base at water level.
In Summary
Key Idea
• If a problem involves calculating a side length or an angle measure, try
to represent the problem with a model that includes right triangles.
If possible, solve the right triangles using the primary trigonometric ratios.
Need to Know
B
• To calculate the area of a triangle, use the sine ratio to determine
the height. For example, suppose that you know a, b, and ∠ C in
h
the triangle at the right. To calculate the height, you can use
sin C =
NEL
A
a
C
b
1
h
, so h = a (sin C). Area of the triangle ⫽ ⫻ b ⫻ a (sin C).
a
2
Chapter 7
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CHECK Your Understanding
1. Isabelle is flying a kite on a windy day. When the kite is 15 m above
ground, it makes an angle of 50° with the horizontal. If Isabelle is
holding the string 1 m above the ground, how much string has she
released? Round your answer to the nearest metre.
15 m
2. Bill was climbing a 6.0 m ladder, which was placed against a wall
at a 76° angle. He dropped one of his tools directly below the ladder.
The tool landed 0.8 m from the base of the ladder. How far from
the top of the ladder was Bill?
50°
1m
6.0 m
76°
0.8 m
3. A guy wire is attached to a cellphone tower as shown at the left.
24 m
30 m
The guy wire is 30 m long, and the cellphone tower is 24 m high.
Determine the angle that is formed by the guy wire and the ground.
?
PRACTISING
4. A tree that is 9.5 m tall casts a shadow that is 3.8 m long.
K
What is the angle of elevation of the Sun?
5. The rise of a rafter drops by 3 units for every 5 units of run.
Determine the angle of depression of the rafter.
6. A building code states that a set of stairs cannot rise more than 72 cm
for each 100 cm of run. What is the maximum angle at which
the stairs can rise?
Career Connection
Jobs in construction include
designer, engineer, architect,
project manager, carpenter,
mason, electrician, plumber,
and welder.
412
7. A contractor is laying a drainage pipe. For every 3.0 m of horizontal
pipe, there must be a 2.5 cm drop in height. At what angle should
the contractor lay the pipe? Round your answer to the nearest tenth
of a degree.
7.6 Solving Right Triangle Problems
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7.6
8. Firefighters dig a triangular trench around a forest fire to prevent the
fire from spreading. Two of the trenches are 800 m long and 650 m
long. The angle between them is 30°. Determine the area that
is enclosed by these trenches.
9. A Mayan pyramid at Chichén-Itzá has stairs that rise about 64 cm for
every 71 cm of run. At what angle do these stairs rise?
10. After 1 h, an airplane has travelled 350 km. Strong winds, however,
have caused the plane to be 48 km west of its planned flight path.
By how many degrees is the airplane off its planned flight path?
History Connection
actual
flight
path
␪
Chichén-Itzá, in the Yucatan
peninsula of Mexico, was part
of the Mayan civilization. The
pyramid called El Castillo, or
the castle, is a square-based
structure with four staircases
and nine terraces.
planned
flight
path
11. Angles were measured from two points on opposite sides of a tree,
A
as shown. How tall is the tree?
35˚
30˚
65 m
12. Determine the angle between the line y =
3
x + 4 and the x-axis.
2
13. A bridge is going to be built across a river. To determine the width of
T
the river, a surveyor on one bank sights the top of a pole, which is 3 m
high, on the opposite bank. His optical device is mounted 1.2 m above
the ground. The angle of elevation to the top of the pole is 8.5°. How
wide is the river?
14. Élise drew a diagram of her triangular yard. She wants to cover her
C
yard with sod. Explain how you could calculate the cost, if sod
costs $1.50/m2.
120 m
40°
100 m
NEL
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15. A video camera is mounted on top of a building that is 120 m tall.
The angle of depression from the camera to the base of another
building is 36°. The angle of elevation from the camera to the top
of the same building is 47°.
47°
36°
h
120 m
distance between
buildings
a) How far apart are the two buildings? Round your answer
to the nearest metre.
b) How tall is the building viewed by the camera? Round your
answer to the nearest metre.
16. An isosceles triangle has a height of 12.5 m (measured from the
unequal side) and two equal angles that measure 55°. Determine
the area of the triangle.
17. To photograph a rocket stage separating, Lucien mounts his camera
on a tripod. The tripod can be set to the angle at which the stage will
separate. This is where Lucien needs to aim his lens. He begins by
aiming his camera at the launch pad, which is 1500 m away. The stage
will separate at 20 000 m. At what angle should Lucien set the tripod?
18. Explain the steps you would use to solve a problem that involves
a right triangle model and the use of trigonometry.
Extending
B
8.2 cm
A
19. Each side length of regular pentagon ABCDE is 8.2 cm.
␪
C
a) Calculate the measure of u to the nearest degree.
b) Calculate the length of diagonal AC to the nearest tenth
of a centimetre.
20. Determine the acute angle at which y = 2x - 1 and y = 0.5x + 2
intersect.
E
414
D
7.6 Solving Right Triangle Problems
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Chapter Review
FREQUENTLY ASKED Questions
Q:
A:
• See Lesson 7.4,
Examples 1 to 3.
The primary trigonometric ratios for ∠A in ^ ABC are
opposite
adjacent
opposite
tan A =
sin A =
cos A =
hypotenuse
hypotenuse
adjacent
To calculate an angle or a side using a trigonometric ratio, follow
these steps:
• Label the sides of the triangle relative to either an acute angle
you know or the angle you want to calculate.
• Use the appropriate trigonometric ratio to write an equation
that involves the angle or side you want to calculate.
• Solve your equation.
hypotenuse
How do you know when to use the inverse trigonometric
ratios?
A:
Use sin -1, cos -1, or tan -1 when you need to determine the measure of
an angle and you know the value of a ratio of two sides in a right triangle.
Q:
What strategies can you use to solve a problem that
involves a right triangle model?
A1:
Draw a diagram to model the problem. If you know the measure of one
acute angle and the length of one side, follow these steps:
• Determine the third angle by subtracting the right angle and
the other known angle from 180°.
• Calculate the two unknown side lengths using trigonometric ratios.
Alternatively, calculate one unknown side length using a
trigonometric ratio and solve for the last side length using the
Pythagorean theorem.
NEL
Aid
• Try Chapter Review
Q:
A2:
Study
What are the primary trigonometric ratios, and
how do you use them?
Questions 5 to 10.
B
opposite
A
adjacent
Study
C
Aid
• See Lesson 7.4, Example 3,
and Lesson 7.5, Example 3.
• Try Chapter Review
Questions 5 b), 7, and 8 b).
Study
Aid
• See Lesson 7.5, Examples 1
to 4, and Lesson 7.6,
Examples 1 to 4.
• Try Chapter Review
Questions 11 to 17.
Draw a diagram to model the problem. If you know two side lengths
but neither acute angle, follow these steps:
• Use inverse trigonometric ratios to calculate one of the missing
angles.
• Calculate the third angle by subtracting the angle you found and
the right angle from 180°.
• Calculate the third side using a trigonometric ratio or
the Pythagorean theorem.
Chapter 7
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PRACTICE Questions
Lesson 7.1
4. Brett needs to support a radio tower with guy
1. Determine whether these triangles are similar. If
they are similar, write a proportion statement
and determine the scale factor.
o
o
9.00
4
wires. Each guy wire must run from the top
of the tower to its own anchor 9.00 m from
the base of the tower. When the tower casts
a shadow that is 9.00 m long, Brett’s shadow
is 0.60 m long. Brett is 1.85 m tall. What
is the length of each guy wire that Brett needs?
11.25
5
3
6.75
Lesson 7.2
2. State whether the triangles in the diagram are
similar. Then determine p.
1.85 m
A
0.60 m
6 cm
9.00 m
Lesson 7.4
5. a) Determine the three primary trigonometric
C
B
p
D
ratios for ∠ A.
5 cm
4 cm
4m
8m
E
3. Calculate the heights of the two ramp supports,
x and y. Round your answers to the nearest
tenth of a metre.
A
b) Calculate the measure of ∠ A to the nearest
degree.
6. Determine x to one decimal place.
a) tan 46° =
x
14.2
b) cos 29° =
17.3
x
Lesson 7.5
7. ABCD is a rectangle
1.3 m
x
0.9 m
416
y
1.2 m
Chapter Review
1.7 m
with AB = 15 cm
and BC = 10 cm.
What is the measure
of ∠ BAC to the
nearest degree?
A
B
D
C
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Chapter Review
8. In ^PQR, ∠ R = 90° and p ⫽ 12.0 cm.
a) Determine r, when ∠ Q = 53°.
b) Determine ∠ P, when q ⫽ 16.5 cm.
9. Solve this triangle.
9 cm ␪
a
18°
b
10. Maria needs to load cars onto a transport
truck. She is planning to drive up a ramp, onto
the truck bed. The truck bed is 1.5 m high,
and the maximum angle of the slope of the
ramp is 35°.
a) How far is the rear of the truck from the
point where the ramp touches the ground?
b) How long should the ramp be? Round your
answer to one decimal place.
14. Two watch towers at an historic fort are located
375 m apart. The first tower is 14 m tall, and
the second tower is 30 m tall.
a) What is the angle of depression from the
top of the second tower to the top of the
first tower?
b) The guards in the towers simultaneously
spot a suspicious car parked between the
towers. The angle of depression from the
lower tower to the car is 7.7°. The angle of
depression from the higher tower is 6.3°.
Which guard is closer to the car? Explain
how you know.
15. Calculate the length of AB using the
information provided. Show all your steps.
A
Lesson 7.6
11. A search-and-rescue airplane is flying at an
altitude of 1200 m toward a disabled ship. The
pilot notes that the angle of depression to the
ship is 12°. How much farther does the airplane
have to fly to end up directly above the ship?
B
20°
20°
100 m
C
D
16. A swimmer observes that from point A, the
12°
1200 m
12. The angle of elevation from the top of a 16 m
building to the top of a second building is 48°.
The buildings are 30 m apart. What is the
height of the taller building?
angle of elevation to the top of a cliff at
point D is 30°. When the swimmer swims
toward the cliff for 1.5 min to point B, he
estimates that the angle of elevation to the top
of the cliff is about 45°. If the height of the cliff
is 70.0 m, calculate the distance the swimmer
swam.
D
13. A cyclist pedals his bike 6.5 km up a mountain
road, which has a steady incline. By the time
he has reached the top of the mountain, he has
climbed 1.1 km vertically. Calculate the angle
of elevation of the road.
70.0 m
45°
30°
A
B
C
17. A plane takes off in a straight line and travels
along this line for 10 s, when it reaches a height
of 300 m. If the plane is travelling at 60 m/s, at
what angle is the plane ascending?
NEL
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Chapter Self-Test
1. Determine the indicated side lengths in the triangles.
Checklist
✔ Questions 2 and 5: Did you
visualize or sketch a
diagram that represents
the information accurately?
11.80
✔ Questions 8 and 9: Did you
reflect on the relationship
between the given
information and the
questions asked as you
solved the problems?
b
o
c
16.48
o o
10.30
✔ Question 7: Did you
communicate your
thinking with words and a
diagram that connect the
situation with trigonometry?
a
o
2. Two trees cast a shadow when the Sun is up. The shadow of one tree
is 12.1 m long. The shadow of the other tree is 7.6 m long. If the
shorter tree is 5.8 m tall, determine the height of the taller tree.
Round your answer to the nearest tenth of a metre.
3. Determine each unknown value. Round your answer to one decimal
place.
x
5
13
b) cos 43° =
y
a) sin 28° =
c) tan A = 7.1154
d) cos B =
7
9
4. Determine the length of the indicated side or the measure
of the indicated angle.
a)
73°
a
50 cm
b)
15 cm
38 cm
␪
5. Solve each triangle.
a) In ^ ABC, ∠ A = 90°, ∠B = 14°, and b ⫽ 5.3 cm.
b) In ^ DEF, ∠ F = 90°, d ⫽ 7.8 mm, and e ⫽ 6.9 mm.
1.20 m
6. A ramp has an angle of elevation of 4.8° and a rise of 1.20 m, as shown
at the left. How long is the ramp and what is its run? Round your
answers to the nearest hundredth of a metre.
4.8°
7. Surveyors need to determine the width of a river. Explain how they can
do this without crossing the river. Use a diagram to illustrate your answer.
8. Jane is on the fifth floor of an office building 16 m above the ground.
She spots her car and estimates that it is parked 20 m from the base of
the building. Determine the angle of depression to the nearest degree.
9. A pilot who is heading due north spots two forest fires. The fire that is
due east is at an angle of depression of 47°. The fire that is due west is
at an angle of depression of 38°. What is the distance between the two
fires, to the nearest metre, if the altitude of the airplane is 2400 m?
418
Chapter Self-Test
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Chapter Task
?
How can you use these materials to determine the height
of your school?
A.
Determine how you can make a clinometer using the protractor,
drinking straw, string, clear tape, and bolt. Make drawings of your
design. Ask a classmate to review your drawings and suggest changes.
B.
Assemble your clinometer. Use it to measure the angle of elevation
to an object whose height you know or can measure.
This Friday!
Suppose that your school is about to have its annual
“Spring Has Sprung” concert. To advertise the
concert, the student council wants to make a banner.
The banner will hang from the roof of the school,
down to the ground. No one on the student council
knows the height of the school. You say that you can
calculate the height, using only these materials:
• protractor
• drinking straw
• string
• clear tape
• bolt
• tape measure
Spring has sprung concert
What’s the Height of Your School?
Task
Checklist
✔ Did you include a diagram
of your clinometer and
an explanation of how
it works?
C.
Test the accuracy of your clinometer using trigonometry. If necessary,
move the string directly across from 90° on the protractor.
D.
Decide on the site where you will determine the angle of elevation
to the roof of your school. Use the tape measure to measure
the distance from this site to the base of the wall of your school.
✔ Did you include a diagram
E.
Which trigonometric ratio will you use to calculate the height?
Compare your answer with your classmates’ answers. Suggest reasons
for any differences.
✔ Did you explain the process
F.
NEL
Prepare a report that explains how your clinometer works and
how you used it to calculate the height of your school.
to show how you
determined the height
of your school?
you used to determine this
height?
✔ Did you clearly summarize
your procedure and results?
Chapter 7
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Aurora
WhitchurchStouffville
9
400
404
Orangeville
407
Richmond Hill
Caledon
Pickering
Markham
Bolton
25
7
Vaughan
10
11
Ajax
401
24
Brampton
TORONTO
Georgetown
25
Mississauga
Guelph
Milton
L
e
ak
O
a
nt
rio
401
6
Oakville
407
Cambridge
54 km
403
6
?
Burlington
8
63°
Brantford
Niagara-onthe-Lake
Grimsby
6
'''''
QEW
St. Catharines
406
20
'''''
Stony
Creek
403
420
43 km
Hamilton
24
U.S.A.
'''''''''' '''''
5
Dundas
Niagara Falls
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Chapter
8
Acute Triangle
Trigonometry
GOALS
You will be able to
• Develop and use the sine law
to determine side lengths and
angle measures in acute triangles
• Develop and use the cosine law
to determine side lengths and
angle measures in acute triangles
• Solve problems that can be modelled
using acute triangles
? Commercial ships transport goods
from city to city around the Great
Lakes.
Why can you not use a primary
trigonometric ratio to directly
calculate the distance by ship
from St. Catharines to Toronto?
NEL
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Getting Started
WORDS YOU NEED to Know
1. Complete each sentence using one of the terms at the left.
a) A triangle in which each interior angle is less than 90° is called
_________.
b) A triangle that contains a 90° angle is called _________.
c) The longest side in a right triangle is called the _________.
adjacent
d) In a right triangle, the ratio
is called _________.
hypotenuse
e) The __________________ describes the relationship between
the three sides in a right triangle.
opposite
f ) In a right triangle, the ratio
is called _________.
hypotenuse
i) sine
ii) an acute triangle
iii) a right triangle
iv) cosine
v) hypotenuse
vi) Pythagorean theorem
SKILLS AND CONCEPTS You Need
Study
Aid
• For more help and practice,
see Appendix A-15.
Angle Relationships
The properties of triangles, including the relationships between angles
formed by transversals and parallel lines, can be used to determine
unknown angle measures.
Alternate angles are equal:
∠3 = ∠6 and ∠4 = ∠5
F
1 2
3 4
5 6
7 8
A
C
B
Corresponding angles are equal:
∠1 = ∠5, ∠2 = ∠ 6,
∠3 = ∠7, and ∠4 = ∠8
D
E
Co-interior angles are supplementary:
∠4 + ∠6 = 180°
∠3 + ∠ 5 = 180°
EXAMPLE
Determine all the unknown angle measures. Explain your reasons.
H
F
G
E
110°
A
422
Getting Started
B
C
D
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Getting Started
Solution
Angle Measure
Reason
∠FCB = 180° - ∠FCD
= 180° - 110°
= 70°
∠ FCB and ∠FCD are supplementary
angles.
∠FBC = ∠FCB
= 70°
FB = FC, so ^FBC is isosceles.
∠BFC = 180° - ∠FCB - ∠FBC
= 180° - 70° - 70°
= 40°
Sum of the interior angles
of a triangle is 180°.
∠FBA = 180° - ∠FBC
= 180° - 70°
= 110°
∠ FBA and ∠FBC are supplementary
angles.
∠EFC = ∠FCB
= 70°
GE || AD, so alternate angles are equal.
∠GFB = ∠FBC
= 70°
GE || AD, so alternate angles are equal.
∠HFE = ∠FCD
= 110°
GE || AD, so corresponding angles
are equal.
∠HFG = 180° - ∠HFE
= 180° - 110°
= 70°
∠HFG and ∠HFE are supplementary
angles.
2. Determine the measures of the indicated angles in each diagram.
a)
c)
g
c
100°
a
b
f
35°
d
h
e
30°
b)
75°
i
25°
k
j
d)
70°
NEL
55°
l
m
n
Chapter 8
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PRACTICE
Aid
Study
5/12/09
3. Which is the longest side and which is the shortest side in each triangle?
• For help, see the Review of
Essential Skills and
Knowledge Appendix.
Question
Appendix
3
A-15
6
a) A
b)
D
65°
48°
50°
B
A-14
E
C
F
4. Which is the greatest angle and which is the least angle in each triangle?
a)
b)
A
D
6 cm
7 cm
4 cm
E
B
3 cm
8 cm
F
C
5. Determine the value of each trigonometric ratio to four decimal places.
a) sin 55°
b) cos 24°
c) cos 82°
d) sin 37°
6. Solve.
x
5
x
c) sin 30° =
=
3
12
12
25
36
9
b)
d) cos 60° =
=
x
x
2
7. Determine the measure of ∠ A to the nearest degree.
3
a) sin A = 0.5
c) sin A =
4
5
b) cos A = 0.5
d) cos A =
8
8. A 3 m board is leaning against a vertical wall. If the base of the board
is placed 1 m from the wall, determine the measure of the angle that
the board makes with the floor.
9. a) Is ^ ABG ' ^DCG in the diagram at the left? Explain how
you know.
AB
b) Write the ratios that are equivalent to
.
DC
10. Belinda claims that the value of any primary trigonometric ratio
in a right triangle will always be less than 1. Do you agree or disagree?
Justify your decision.
a)
A
B
G
C
D
424
Getting Started
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Getting Started
APPLYING What You Know
Soccer Trigonometry
YOU WILL NEED
• ruler
• protractor
Marco is about to take a shot in front of a soccer net. He estimates
that his current position
• is 5.5 m from the left goalpost and 6.5 m from the right goalpost
• forms a 75° angle with the two goalposts
goal
5.5 m
6.5 m
75°
Marco
?
How can you use these measurements to calculate the width
of the soccer net?
A.
Does Marco’s position form a right triangle with the goalposts?
B.
Can a primary trigonometric ratio be used to calculate the width
of the net directly? Explain.
C.
Copy the triangle in the diagram above. Add a line so that you can
calculate the height of the triangle using a primary trigonometric ratio.
D.
Calculate the height of the triangle.
E.
Create and describe a plan that will allow you to calculate the width
of the soccer net using the two right triangles you created.
F.
Carry out your plan to calculate the width of the soccer net.
NEL
Chapter 8
425
8.1
Exploring the Sine Law
YOU WILL NEED
GOAL
• dynamic geometry software,
Explore the relationship between each side in an acute triangle
and the sine of its opposite angle.
or ruler and protractor
EXPLORE the Math
The primary trigonometric ratios––sine, cosine, and tangent––are defined
for right triangles.
Tech
Support
?
What is the relationship between a side and the sine
of the angle that is opposite this side in an acute triangle?
A.
Construct an acute triangle, ^ ABC, and measure all its angles and
sides to one decimal place. Record the measurements in a table like
the one below.
For help using dynamic
geometry software to
construct a triangle, measure
its angles and sides, and
calculate, see Appendix B-25,
B-26, B-29, and B-28.
426
Angle
Side
Sine
length of opposite side
sin (angle)
∠A ⫽
a⫽
sin A ⫽
a
=
sin A
∠B ⫽
b⫽
sin B ⫽
b
=
sin B
∠C ⫽
c⫽
sin C ⫽
c
=
sin C
B.
Determine the sine of each angle. Calculate the ratio
length of opposite side
for each angle in the triangle.
sin (angle)
Record these values in your table.
C.
What do you notice about the value of the ratio for each angle?
D.
Create a different acute triangle, and repeat part B. What do you
notice?
E.
Create several more acute triangles, and repeat parts B and C.
F.
Create a right triangle, and repeat part B. What do you notice?
G.
Investigate whether replacing the sine ratio with the cosine or tangent
ratio gives the same results.
8.1 Exploring the Sine Law
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8.1
H.
Explain what you have discovered
i) in words
ii) with a mathematical relationship
Reflecting
I.
Suggest an appropriate name for the relationship you have discovered.
J.
Does this relationship guarantee that if you know the measurements of
an angle and the opposite side in a triangle, as well as the measurement
of one other side or angle, you can calculate the measurements of the
other angles and sides? Explain.
In Summary
Key Idea
length of opposite side
is the same for all three angle –side
sin (angle)
pairs in an acute triangle.
• The ratio
C
Need to Know
• In an acute triangle, ^ABC,
a
b
c
=
=
sin A
sin B
sin C
• This relationship is also true for right triangles.
b
a
A
c
B
FURTHER Your Understanding
1. For each acute triangle,
i) copy the triangle and label the sides using lower-case letters
ii) write the ratios that are equivalent
X
a) L
b)
N
Y
M
Z
2. Solve for the unknown side length or angle measure. Round
your answer to one decimal place.
6.0
w
8.0
10.0
a)
c)
=
=
sin 50°
sin 60°
sin M
sin 72°
b)
k
9.5
=
sin 43°
sin 85°
d)
12.5
14.0
=
sin Y
sin 88°
3. Matt claims that if a and b are adjacent sides in an acute triangle, then
a sin B ⫽ b sin A. Do you agree or disagree? Justify your decision.
4. If you want to calculate an unknown side length or angle measure in an
acute triangle, what is the minimum information that you must have?
NEL
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Applying the Sine Law
YOU WILL NEED
GOAL
• ruler
Use the sine law to calculate unknown side lengths and
angle measures in acute triangles.
LEARN ABOUT the Math
sine law
in any acute triangle,
b
c
a
=
=
sin A
sin B
sin C
In Lesson 8.1, you discovered the sine law for acute triangles. Can you be
sure that the sine law is true for every acute triangle?
length of opposite side
is
sin (angle)
the same for all three angle-side pairs in any acute triangle?
How can you show that the ratio
?
C
b
a
A
EXAMPLE 1
c
B
Proving the sine law for acute triangles
Show that the sine law is true for all acute triangles.
Ben’s Solution
A
c
B
b
D
In ^ ABD,
AD
sin B =
c
c sin B = AD
428
8.2 Applying the Sine Law
I drew an acute triangle. Since
the primary trigonometric ratios
are used for right triangles only, I
drew a perpendicular height, AD,
from A to BC to form two right
triangles, ^ ABD and ^ ACD.
C
In ^ ACD,
AD
sin C =
b
Since the sine law involves the
sine of the angles in ^ ABC,
I wrote equations for the sines
of ∠ B and ∠ C in the two right
triangles using the ratio
opposite
.
hypotenuse
b sin C = AD
I knew that AD was the opposite
side in both right triangles, so
I used the sine ratio to describe
AD in ^ ABD and ^ ACD.
I thought this would allow me
to relate the two triangles.
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8.2
Since the expressions c sin B and
b sin C both describe AD, I set
them equal.
c sin B ⫽ b sin C
c sin B
= b
sin C
I wanted to write each side of the
equation as a ratio with information
about only one triangle. I divided
both sides of the equation by sin C
and then by sin B.
c
b
=
sin C
sin B
A
E
c
a
B
In ^ ABE ,
BE
sin A =
c
c sin A = BE
I wanted to show that the ratios
c
b
a
and
were equal to
.
sin C
sin B
sin A
C
In ^ CBE ,
BE
sin C =
a
a sin C = BE
c sin A ⫽ a sin C
a sin C
c =
sin A
a
c
=
sin C
sin A
a
b
c
=
=
sin A
sin B
sin C
To do this, I needed to divide ^ ABC
differently so that I could use ∠ A. I
reasoned that if I drew a height BE
from B to AC, I could follow the
same steps I used for height AD.
Since the expressions c sin A and
a sin C both describe BE, I set them
equal. Then I divided both sides
of the equation by sin A and then
by sin C.
Since
b
a
and
are equal to
sin B
sin A
c
, all three ratios must be equal.
sin C
Reflecting
A.
Why did Ben need to draw line segment AD perpendicular to side BC ?
B.
If Ben drew a perpendicular line segment from vertex C to side AB,
which pair of ratios in the sine law do you think he could show
are equal?
C.
Why does it make sense that the sine law can also be written
sin A sin B sin C
in the form
⫽
⫽
?
c
a
b
NEL
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APPLY the Math
EXAMPLE 2
Selecting a sine law strategy to calculate the length of a side
A
Determine the length of AC.
80°
55°
B
45°
12.0 m
C
Elizabeth’s Solution
A
I named the sides of the triangle using lower-case
letters that correspond to the opposite angles. Then
I identified the measures I didn’t know.
80°
c
b
55°
45°
a ⫽ 12.0 m
B
C
b
a
=
sin A
sin B
12.0
b
=
sin 80°
sin 55°
sin 55° a
12.0
b
b = sin 55° a
b
sin 80°
sin 55°
12.0
b = b
sin 55° a
sin 80°
12.0
#
b = b
0.8192 a
0.9848
#
9.98 = b
The length of AC is about 10.0 m.
EXAMPLE 3
Since the triangle does not contain a right angle,
I couldn’t use the primary trigonometric ratios.
I realized that I could use the sine law if I knew an
opposite side-angle pair, plus one more side or angle
in the triangle. I knew a and ∠A and I wanted to
know b, so I related a, b, sin A, and sin B. I could
sin B
sin A
=
have used
, but I decided to use
a
b
b
a
=
since I had to solve for b. This meant
sin A
sin B
that b was the numerator and I could multiply both
sides by sin 55° to solve for b.
It made sense that b is shorter than a, since ∠ B
is less than ∠ A.
Selecting a sine law strategy to calculate the measure of an angle
In ^ DST , ∠ D ⫽ 47°, d ⫽ 78 cm, and s ⫽ 106 cm. Determine
the measure of ∠ S.
430
8.2 Applying the Sine Law
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8.2
Phil’s Solution
S
d ⫽ 78 cm
t
D
47°
s ⫽ 106 cm
T
sin D
sin S
=
s
d
sin 47°
sin S
=
106
78
106 a
sin S
sin 47°
b = 106 a
b
106
78
sin 47°
b
sin S = 106 a
78
sin S ⬟ 0.9939
∠ S = sin - 1(0.9939)
#
∠ S = 83.7°
I drew a diagram. I knew that ∠ S was greater than
∠ D since s 7 d. I couldn’t assume that the triangle
contained a right angle, so I couldn’t use the
primary trigonometric ratios.
I knew s, d, and ∠ D, and I wanted to determine
the measure of ∠ S. So, I used the sine law that
included these four quantities. I used the
proportion with sin S and sin D in the numerators
to make the equation easier to solve for sin S.
To determine ∠ S, I calculated the inverse
of the sine ratio.
∠ S is about 84°.
EXAMPLE 4
Solving a problem using the sine law
The roof of a new house must be built to exact specifications so
that solar panels can be installed. The long rafters at the front of
the house must be inclined at an angle of 26° to the horizontal
beam. The short rafters at the back of the house must be inclined
at an angle of 66°. The house is 15.3 m wide. Determine the
length of the long rafters.
long
rafters
26°
short
rafters
beam
66°
15.3 m
Taylor’s Solution
C
b
66°
26°
A
a
c ⫽ 15.3 m
B
∠ C = 180° - 26° - 66°
= 88°
NEL
I drew an acute triangle to model the situation. The
long rafters were opposite ∠ B, the 66° angle, so I
labelled them side b. I knew that my diagram was
correct, and the long rafters were side b, since
66° 7 26°.
Since I knew length c, I had to use ratios that
involved ∠ B and ∠ C. So, I had to determine the
measure of ∠ C. I knew that the angles in a triangle
add up to 180°.
Chapter 8
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c
b
=
sin B
sin C
15.3
b
=
sin 66°
sin 88°
sin 66° a
Page 432
Since I needed to determine side b, I used the sine
law to write a proportion with the sides in the
numerator.
15.3
b
b = sin 66° a
b
sin 66°
sin 88°
15.3
b
b = sin 66° a
sin 88°
#
b = 13.99
I solved for b.
Since ∠ B is less than ∠ C, it makes sense that b is
shorter than c.
The long rafters are about 14.0 m long.
In Summary
Key Idea
• The sine law can be used to determine unknown side lengths or
angle measures in some acute triangles.
Need to Know
• To use the sine law to determine a side length or angle measure,
follow these steps:
• Determine the ratio of the sine of a known angle measure and
a known side length.
• Create an equivalent ratio using the unknown side length and the
measure of its opposite angle, or the sine of the unknown angle
measure and the length of its opposite side.
• Equate the ratios you created, and solve.
• You can use the sine law to solve a problem modelled by an acute
triangle if you know the measurements of
• two sides and the angle that is opposite one of these sides
• two angles and any side
• An acute triangle can be divided into right triangles. The proof of the
sine law involves writing proportions that compare corresponding sides
in these right triangles.
CHECK Your Understanding
1. Write three equivalent ratios using the sides
S
and angles in the triangle at the right.
r
Q
432
8.2 Applying the Sine Law
q
s
R
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8.2
2. Determine the indicated measures to one decimal place.
a)
b)
A
b
29.5 cm
37.1 cm
27.2 cm
43°
72°
A
C
␪
44°
C
B
B
PRACTISING
3. Determine the indicated side lengths and angle measures.
X
a)
c)
D
S
y
88°
3.0 cm
53°
e)
22.5 cm
␣
Y
Z
d
b) A
40.0 cm
B
60°
d)
f)
N
L
a
b
6.5 cm
Q
E
40°
␪
50°
68°
F
5.0 cm
25°
␪
29°
45.2 cm
24.4 cm
J
2.9 m
␣
6.7 m
80°
M
R
K
␪
j
L
C
4. Scott is a naturalist. He is studying the
A
effects of acid rain on fish populations in
different lakes. As part of his research, he
needs to know the length of Lake Lebarge.
Scott makes the measurements shown.
How long is Lake Lebarge?
A
Lake
Lebarge
52°
B
48°
4340 m
C
5. Draw a labelled diagram for each triangle. Then calculate the required
side length or angle measure.
a) In ^SUN , n ⫽ 58 cm, ∠ N = 38°, and ∠U = 72°. Determine
the length of side u.
b) In ^PQR , ∠ R = 73°, ∠ Q = 32°, and r = 23 cm. Determine
the length of side q.
c) In ^TAM , t = 8 cm, m = 6 cm, and ∠T = 65°. Determine
the measure of ∠M .
d) In ^WXY, w = 12.0 cm, y = 10.5 cm, and ∠W = 60°.
Determine the measure of ∠ Y.
6. In ^CAT, ∠ C = 32°, ∠ T = 81°, and c = 24.1 m.
K
NEL
Solve the triangle.
Environment Connection
Acidic lakes cannot support
the variety of life in healthy
lakes. Clams and crayfish are
the first to disappear, followed
by other species of fish.
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7. The short sides of a parallelogram are both 12.0 cm. The acute angles
of the parallelogram are 65°, and the short diagonal is 15.0 cm.
Determine the length of the long sides of the parallelogram. Round
your answer to the nearest tenth of a centimetre.
8. An architect designed a house that is 12.0 m wide. The rafters that
hold up the roof are equal in length and meet at an angle of 70°,
as shown at the left. The rafters extend 0.3 m beyond the supporting
wall. How long are the rafters?
70°
0.3 m
9. A telephone pole is supported by two wires on opposite sides.
12.0 m
T
10. In ^PQR, ∠ Q = 90°, r = 6, and p ⫽ 8. Explain two different ways
to calculate the measure of ∠ P.
210 m
60°
At the top of the pole, the wires form an angle of 60°. On the ground,
the ends of the wires are 15.0 m apart. One wire makes a 45° angle
with the ground. How long are the wires, and how tall is the pole?
11. A bridge across a gorge is 210 m long, as shown in the diagram at the
75°
left. The walls of the gorge make angles of 60° and 75° with the
bridge. Determine the depth of the gorge to the nearest metre.
12. Use the sine law to help you describe each situation.
C
13. Jim says that the sine law cannot be used to determine the length
C
of side c in ^ ABC at the left. Do you agree or disagree? Explain.
70°
3.2 m
a) Three pieces of information allow you to solve for all the unknown
side lengths and angle measures in a triangle.
b) Three pieces of information do not allow you to solve a triangle.
14. Suppose that you know the length of side p in ^ PQR, as well as
the measures of ∠ P and ∠ Q. What other sides and angles could you
3.0 m
calculate? Explain how you would determine these measurements.
A
c
B
Extending
15. In ^ ABC, ∠A = 58°, ∠ C = 74°, and b = 6. Calculate the area
of ^ ABC to one decimal place.
16. An isosceles triangle has two sides that are 10 cm long and two angles
that measure 50°. A line segment bisects one of the 50° angles and
ends at the opposite side. Determine the length of the line segment.
17. Use the sine law to write a ratio that is equivalent to each expression
for ^ABC .
a
a)
sin A
434
8.2 Applying the Sine Law
b)
sin A
sin B
c)
a
c
d)
a sin C
c sin A
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Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
What is the sine law, and what is it used for?
A:
The sine law describes the relationship between sides and their
opposite angles in a triangle. According to the sine law, the ratio
length of opposite side
is the same for all three angle-side pairs
sin (angle)
in a triangle.
Aid
• See Lessons 8.1 and 8.2.
• Try Mid-Chapter Review
Questions 1 to 3.
C
In ^ ABC,
b
c
a
=
=
sin A
sin B
sin C
or
sin A
sin B
sin C
=
=
c
a
b
Study
b
a
A
c
B
The sine law can be used to determine unknown side lengths and
angle measures in acute and right triangles.
Q:
When can you use the sine law?
A:
You can use the sine law if you know any three of these four
measurements: two side lengths and the measures of their opposite
angles. The sine law will allow you to calculate the fourth side length or
angle measure. If you know any two angle measures in a triangle, you
can calculate the third angle measure.
Study
Aid
• See Lesson 8.2, Examples 2
to 4.
• Try Mid-Chapter Review
Questions 4 to 9.
EXAMPLE
Can the sine law be used to determine the length of AB in the triangle
at the right? Explain.
A
52°
Solution
Yes.
Side AB (or c) is opposite ∠ C,
which equals 72°.
Side AC (or b) equals 500 m and
is opposite ∠ B.
∠ B = 180° - 52° - 72°
= 56°
NEL
Solving for c in this proportion gives
the length of AB.
c
500
=
sin 72°
sin 56°
500
c = sin 72° a
b
sin 56°
c ⬟ 573.6
500 m
72°
C
B
The length of AB is about 574 m.
Chapter 8
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PRACTICE Questions
Lesson 8.1
6. In ^ XYZ, the values of x and z are known. What
1. What relationship(s) does the sine law describe
in the acute triangle XYZ ?
2. Why are you more likely to use the sine law
for acute triangles than for right triangles?
3. ^DEF is an acute triangle. Nazir claims that
f
d
. Do you agree or disagree?
=
sin F
sin D
Explain.
additional information do you need to know if
you want to use the sine law to solve the triangle?
7. Two fire towers in a park are 3.4 km apart.
When the park rangers on duty spot a fire, they
can locate the fire by measuring the angle
between the fire and the other tower. A fire
is located 53° from one tower and 65° from
the other tower.
Lesson 8.2
4. a) Determine the measure of u and the length
of side x.
4.5 cm
4.0 cm
␪
50°
53°
65°
3.4 km
x
b) Determine the measure of u and the lengths
of sides x and y.
␪
a) Which tower is closer to the fire?
b) Determine the distance from the closest
tower to the fire.
8. As Chloe and Ivan canoe across a lake, they
x
y
55°
63°
10.5 cm
notice a campsite ahead at an angle of 22° to
the left of their direction of paddling. After
continuing to paddle in the same direction for
800 m, the campsite is behind them at an angle
of 110° to their direction of paddling. How far
away is the campsite at the second sighting?
5. In ^ ABC, ∠ A = 70°, ∠ B = 50°, and
a = 15 cm. Solve ^ ABC.
B
50°
15 cm
9. Calculate the perimeter of an isosceles triangle
70°
A
C
436
Mid-Chapter Review
with
a) a base of 25 cm and only one angle of 50°
b) a base of 30 cm and equal angles of 55°
NEL
8.3
Exploring the Cosine Law
YOU WILL NEED
• dynamic geometry software
GOAL
Explore the relationship between side lengths and angle measures
in a triangle using the cosines of angles.
EXPLORE the Math
Sometimes, you cannot use the sine law to determine an unknown side
length or angle measure in an acute triangle. This occurs when you do not
have enough information to write a ratio comparing a side length and its
opposite angle. For example, consider these two triangles:
C
F
66°
2.6 m
D
3.6 m
E
B
c
?
How can the Pythagorean theorem be extended to relate
the sides and angles in these two triangles?
A.
Use dynamic geometry software to construct any acute triangle. Label the
vertices A, B, and C and the sides a, b, and c, as in the triangle at the right.
Measure all three interior angles and all three sides.
C.
Drag a vertex until ∠ C ⫽ 90°. State the Pythagorean relationship
for this triangle.
D.
Calculate.
i) a2 + b2
iii) a2 + b2 - c2
ii) c2
iv) 2 ab cos C
Record your results in a table like the one shown.
1
2
a
b
c
∠C
90°
c2
a2 ⴙ b2
a2 ⴙ b2 ⴚ c 2
C
a
b
B.
Triangle
Support
For help using dynamic
geometry software to
construct a triangle, measure
its angles and sides, and
calculate, see Appendix B-25,
B-26, B-29, and B-28.
2.5 m
3.1 m
3.2 m
A
Tech
2 ab cos C
B
c
A
Communication
Tip
2 ab cos C is a product
in which four terms are
multiplied: (2)(a)(b)(cos C ).
3
4
5
NEL
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E.
Drag vertex C away from side AB to create a new acute triangle.
Repeat part D for this triangle.
F.
What do you notice about your results for a2 + b2 - c2 and 2 ab cos C ?
G.
Drag vertex C to at least five other positions. Repeat part D for each
new triangle you create.
H.
Based on your observations, what can you conclude about
the relationship among a2, b2, and c2 in an acute triangle?
Reflecting
I.
When ∠ C ⫽ 90°, what happens to the value of 2 ab cos C
in the cosine law ? Why does this happen?
in any acute triangle,
c2 = a2 + b2 - 2 ab cos C
J.
How does the measure of ∠ C affect the value of 2 ab cos C ?
C
K.
Explain how the cosine law could be used to relate
i) the value of a2 to the value of b2 + c2
ii) the value of b2 to the value of a2 + c2
cosine law
b
a
A
c
In Summary
B
Key Idea
• The cosine law is an extension of the Pythagorean theorem to triangles
with no right angle.
Need to Know
• The cosine law states that for any ^ ABC,
a2 = b 2 + c 2 - 2 bc cos A
b 2 = a2 + c 2 - 2 ac cos B
c 2 = a2 + b 2 - 2 ab cos C
C
b
a
A
c
B
FURTHER Your Understanding
1. a) Sketch ^LMN at the left, and label each side using lower-case letters.
L
b) Relate each side length in ^LMN to the cosine of its opposite
angle and the lengths of the other two sides.
N
2. Sketch each triangle, and then solve for the unknown side length or
angle measure.
a) w 2 = 152 + 162 - 2(15)(16)cos 75°
b) k 2 = 322 + 352 - 2(32)(35)cos 50°
c) 482 = 462 + 452 - 2(46)(45)cos Y
d) 132 = 172 + 152 - 2(17)(15)cos G
M
438
8.3 Exploring the Cosine Law
NEL
8.3
3. Martina used algebra to write the cosine law as follows:
r 2 = p 2 + q 2 - 2 pq cos R
r 2 - p 2 - q 2 = - 2 pq cos R
- 2 pq cos R
r 2 - p2 - q 2
=
- 2 pq
- 2 pq
2
2
2
p + q - r
= cos R
2 pq
a) Explain the advantage of writing the cosine law this way.
b) Write the cosine law for cos P in terms of sides p, q, and r.
c) Write the cosine law for cos Q in terms of sides p, q, and r.
4. Identify what you need to know about a triangle if you want to use
the cosine law to calculate
a) an unknown side length
b) an unknown angle measure
5. Express the cosine law in words to describe the relationship between
the three sides in an acute triangle and the cosine of one angle.
YOU WILL NEED
• dynamic geometry software
Curious Math
The Law of Tangents
You have seen that the sine law and cosine law relate the sines and cosines
of angles in acute and right triangles to the measures of their sides. Does
a similar relationship exist for the tangents of the angles in a triangle?
C
1. Using dynamic geometry software, construct an
acute triangle. Label the angles and sides as shown.
2. Measure all three sides and all three angles.
a
b
A
c
B
a - b
3. Select the lengths of a and b. Determine the value of the ratio
.
a + b
4. Select the measures of ∠ A and ∠ B. Determine the value
of the ratio
tan A 12 (A - B) B
tan A 12 (A + B) B
. What do you notice?
5. Make a conjecture about what the tangent law for triangles might be.
Test your conjecture by dragging one of the vertices to a new position.
Repeat this two more times, using a different vertex each time.
6. Write the law of tangents in terms of
a) ∠ B, ∠ C, and sides b and c
NEL
b) ∠ A, ∠C, and sides a and c
Chapter 8
439
8.3
3. Martina used algebra to write the cosine law as follows:
r 2 = p 2 + q 2 - 2 pq cos R
r 2 - p 2 - q 2 = - 2 pq cos R
- 2 pq cos R
r 2 - p2 - q 2
=
- 2 pq
- 2 pq
2
2
2
p + q - r
= cos R
2 pq
a) Explain the advantage of writing the cosine law this way.
b) Write the cosine law for cos P in terms of sides p, q, and r.
c) Write the cosine law for cos Q in terms of sides p, q, and r.
4. Identify what you need to know about a triangle if you want to use
the cosine law to calculate
a) an unknown side length
b) an unknown angle measure
5. Express the cosine law in words to describe the relationship between
the three sides in an acute triangle and the cosine of one angle.
YOU WILL NEED
• dynamic geometry software
Curious Math
The Law of Tangents
You have seen that the sine law and cosine law relate the sines and cosines
of angles in acute and right triangles to the measures of their sides. Does
a similar relationship exist for the tangents of the angles in a triangle?
C
1. Using dynamic geometry software, construct an
acute triangle. Label the angles and sides as shown.
2. Measure all three sides and all three angles.
a
b
A
c
B
a - b
3. Select the lengths of a and b. Determine the value of the ratio
.
a + b
4. Select the measures of ∠ A and ∠ B. Determine the value
of the ratio
tan A 12 (A - B) B
tan A 12 (A + B) B
. What do you notice?
5. Make a conjecture about what the tangent law for triangles might be.
Test your conjecture by dragging one of the vertices to a new position.
Repeat this two more times, using a different vertex each time.
6. Write the law of tangents in terms of
a) ∠ B, ∠ C, and sides b and c
NEL
b) ∠ A, ∠C, and sides a and c
Chapter 8
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YOU WILL NEED
10:52 AM
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Applying the Cosine Law
GOAL
• ruler
Use the cosine law to calculate unknown measures of sides and
angles in acute triangles.
LEARN ABOUT the Math
In Lesson 8.3, you discovered the cosine law for acute triangles. Can you be
sure that the cosine law is true for every acute triangle?
?
EXAMPLE 1
How can you show that the cosine law is true for all acute
triangles?
Proving the cosine law for acute triangles
Show that the cosine law is true for all acute triangles.
Heather’s Solution
A
c
b
a
B
I started by drawing an acute triangle ABC.
C
A
c
x
B
h
D
a
The cosine law is just an extension of the Pythagorean
theorem. I thought that I might be able to relate the angles
and sides in 䉭 ABC if I could create right triangles. I drew a
line segment from A to D so that it was perpendicular to BC.
I labelled this line segment h. I used x for BD and y for DC.
b
y
C
c2 = h2 + x2, so h2 = c2 - x2
b2 = h2 + y2, so h2 = b2 - y2
I wrote the Pythagorean theorem for each triangle to
determine two different expressions for h2.
c2 - x2 = b2 - y2
Then I set the two expressions equal.
x = a - y, so
c2 - (a - y)2 = b2 - y2
c2 = (a - y)2 + b2 - y2
c2 = a2 - 2ay + y2 + b2 - y2
c2 = a2 + b2 - 2ay
I didn’t want to include both x and y in the same equation.
I only wanted one of them. So, I substituted x = a - y for x.
I simplified the equation by expanding and collecting like
terms.
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8.4 Applying the Cosine Law
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8.4
The cosine law for this triangle had to include side lengths a, b,
and c, as well as one angle. My equation c2 = a2 + b2 - 2ay
included the three side lengths, but it also included y, which
I didn’t know. My equation did not involve any angles.
y
, so
b
b cos C y
cos C =
I had to write y in terms of one of the angles in the triangle.
Since y is adjacent to ∠ C in 䉭ADC, I decided to write y
in terms of the cosine of ∠ C.
c 2 = a 2 + b 2 - 2ay
c 2 = a 2 + b 2 - 2 ab cos C
I substituted the expression b cos C for y into my equation.
Reflecting
Why did it make sense for Heather to divide the acute triangle
into two right triangles?
Suppose that Heather had substituted a x for y instead of a y for x.
Would her result have been the same? How do you know?
A.
B.
APPLY the Math
EXAMPLE 2
Selecting a cosine law strategy to calculate the length of a side
A
Determine the length of CB.
32 m
58°
40 m
C
B
Justin’s Solution
A
b 32 m
58°
c 40 m
C
a?
B
a 2 = b 2 + c 2 - 2 bc cos A
a 2 = 322 + 402 - 2(32)(40)cos 58°
NEL
I copied the triangle and named the sides using lower-case
letters. Then I identified the measure that I had to
determine. Since the triangle did not contain a right angle,
I couldn’t use primary trigonometric ratios. I couldn’t use
the sine law either, because I didn’t know a side length
and the measure of its opposite angle.
I knew two sides (b and c) and the angle between these
sides ( ∠ A). I had to determine side a, which is opposite
∠ A. The cosine law relates these four measurements, so
I substituted the values I knew into the cosine law.
Chapter 8
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a 2 = 1024 + 1600 - 2560 cos 58°
a 2 = 2624 - 2560 cos 58°
a2 = 1267.41
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I evaluated the right side. Then I calculated the square root.
a = 21267.41
#
a = 35.6
CB is about 36 m.
EXAMPLE 3
Selecting a cosine law strategy to calculate the measure of an angle
The posts of a hockey net are 1.8 m apart. A player tries to score a goal
by shooting the puck along the ice from a point that is 4.3 m from one
goalpost and 4.0 m from the other goalpost. Determine the measure
of the angle that the puck makes with both goalposts.
Darcy’s Solution
A
b 4.3 m
c 1.8 m
C a 4.0 m
B
I drew a diagram to represent the
situation. I couldn’t assume that this
triangle contained a right angle, so I
couldn’t use primary trigonometric
ratios. I didn’t know the measure of an
angle and a side length opposite the
angle, so I couldn’t use the sine law.
c 2 = a 2 + b 2 - 2 ab cos C
1.82 = 4.02 + 4.32 2(4.0)(4.3) cos C
I had to determine the measure of ∠ C.
The cosine law relates the three sides
of a triangle to an angle in the triangle.
I substituted the measures of the sides
in this triangle into the cosine law.
3.24 = 16.00 + 18.49 - 34.40 cos C
3.24 - 16.00 - 18.49 = - 34.40 cos C
31.25 34.40 cos C
- 31.25
= cos C
- 34.40
#
0.9084 = cos C
cos -1(0.9084) = ∠ C
24.7° ⬟ ∠C
I simplified and then solved for cos C.
I used the inverse cosine
to calculate ∠ C.
The puck makes an angle of about 25° with the goalposts.
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8.4
In Summary
Key Idea
• The cosine law can be used to determine an unknown side length or
angle measure in an acute triangle.
Need to Know
• You can use the cosine law to solve a problem that can be modelled
by an acute triangle if you can determine the measurements of
• two sides and the angle between them
• all three sides
• An acute triangle can be divided into smaller right triangles by drawing
a perpendicular line from a vertex to the opposite side. The proof of the
cosine law involves applying the Pythagorean theorem and cosine ratio
to these right triangles.
CHECK Your Understanding
A
1. Suppose that you are given each set of data for ^ ABC at the right.
Can you use the cosine law to determine c? Explain.
a) a 5 cm, ∠ A 52°, ∠ C 43°
b) a 5 cm, b 7 cm, ∠ C 43°
c
B
b
2. a) Determine the length of side x. b) Determine the measure of ∠ P.
X
18 cm
Q
Y
46°
a
5.9 m
C
R
15 cm
2.3 m
x
6.2 m
P
Z
PRACTISING
3. Determine each unknown side length.
a)
b) D
A
F
10.5 cm
13 cm
11 cm
75°
B
NEL
40°
9.5 cm
C
E
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4. Determine the measure of each indicated angle to the nearest degree.
X
a)
b)
P
␪
3.5 cm
3.9 cm
2.9 cm
2.6 cm
␪
Q
2.2 cm
R
2.2 cm
Y
Z
5. Solve each triangle.
K
a) In ^DEF, d 5.0 cm, e 6.5 cm, and ∠ F 65°.
b) In ^PQR, p 6.4 m, q 9.0 m, and ∠ R 80°.
c) In ^LMN, l 5.5 cm, m 4.6 cm, and n 3.3 cm.
d) In ^XYZ, x 5.2 mm, y 4.0 mm, and z 4.5 cm.
6. Determine the perimeter of ^SRT, if ∠ S 60°, r 15 cm, and
t 20 cm.
7. An ice cream company is designing waffle cones to use for serving
17 cm
frozen yogurt. The cross-section of the design has a bottom angle
of 36°. The sides of the cone are 17 cm long. Determine the diameter
of the top of the cone.
17 cm
8. A parallelogram has sides that are 8 cm and 15 cm long. One of the
C
36°
angles in the parallelogram measures 70°. Explain how you could
calculate the length of the shortest diagonal.
9. The pendulum of a grandfather clock is
A
100.0 cm long. When the pendulum
swings from one side to the other side,
the horizontal distance it travels is
9.6 cm, as in the diagram at the right.
Determine the angle through which the
pendulum swings. Round your answer
to the nearest tenth of a degree.
␪
100.0 cm
10. a) A clock has a minute hand that is
20 cm long and an hour hand that
is 12 cm long. Calculate the
distance between the tips of the hands at
i) 2:00
ii) 10:00
b) Discuss your results for part a).
444
8.4 Applying the Cosine Law
100.0 cm
9.6 cm
NEL
8.4
11. The bases in a baseball diamond are 90 ft apart. A player picks up
a ground ball 11 ft from third base, along the line from second base to
third base. Determine the angle that is formed between first base, the
player’s present position, and home plate.
12. Sally makes stained glass windows. Each piece of glass is surrounded
by lead edging. Sally claims that she can create an acute triangle in part
of a window using pieces of lead that are 15 cm, 36 cm, and 60 cm.
Is she correct? Justify your decision.
13. Two drivers leave home at the same time and travel on straight roads
that diverge by 70°. One driver travels at an average speed of
83.0 km/h. The other driver travels at an average speed of
95.0 km/h. How far apart will the two drivers be after 45 min?
a regular decagon to each vertex
is 12 cm. Calculate the area of
the decagon.
12 cm
first
base
home
plate
The first baseball game
recorded in Canada was
played in Beachville, Ontario,
on June 4, 1838.
O
15. Use the triangle at the right to create
a problem that involves side lengths
and interior angles. Then describe
how to determine the length of side d.
third
base
History Connection
14. The distance from the centre, O, of
T
second
base
d
30 m
35°
35 m
Extending
16. An airplane is flying from Montréal to Vancouver. The wind is
blowing from the west at 60 km/h. The airplane flies at an airspeed
of 750 km/h and must stay on a heading of 65° west of north.
a) What heading should the pilot take to compensate for the wind?
b) What is the speed of the airplane relative to the ground?
17. Calculate the perimeter and area of this regular pentagon. O is the
centre of this pentagon.
O
1.5 cm
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Chapter 8
445
8.5
Solving Acute Triangle
Problems
YOU WILL NEED
GOAL
• ruler
Solve problems using the primary trigonometric ratios and
the sine and cosine laws.
LEARN ABOUT the Math
Reid’s hot-air balloon is 750.0 m directly above a highway. When Reid
is looking west, the angle of depression to Exit 85 is 75°. Exit 83 is located
2 km to the east of Exit 85.
␪
75°
750.0 m
?
2 km
Exit 85
Exit 83
EXAMPLE 1
What is the angle of depression to Exit 83 when Reid
is looking east?
Solving a problem using an acute triangle model
Determine the angle of depression, to the nearest degree, from
the balloon to Exit 83.
Vlad’s Solution
75°
x
75°
␪
The ground and the horizontal are parallel, so the
alternate angles are equal. The angle of elevation
to the balloon at Exit 83 equals u, the angle of
depression I needed to determine. The distance
between the exits is 2 km, or 2000 m.
y
750.0 m
␪
2000 m
Exit 85
Exit 83
750.0
x
x sin 75° = 750.0
750.0
x =
sin 75°
#
x = 776.46
sin 75° =
446
8.5 Solving Acute Triangle Problems
I labelled the unknown sides of the large triangle x
and y. If I could determine both x and y, then I
could calculate u using the sine ratio.
In the right triangle that contains the 75° angle, x is
the hypotenuse and the 750.0 m side is the
opposite side. I used the sine ratio to write an
equation. Then I solved for x.
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8.5
y 2 = x 2 + 20002 - 2(x)(2000)cos 75°
y 2 = 776.462 + 20002 - 2(776.46)(2000)cos 75°
y 2 = 602 890.1316 + 4 000 000 - 803 850.543
y 2 = 3 799 039.589
y = 23 799 039.589
#
y = 1949.11
opposite
sin u =
hypotenuse
750.0
sin u =
1949.11
sin u ⬟ 0.3848
u = sin - 1(0.3848)
#
u = 22.6°
I now knew the lengths of two sides in the large
acute triangle and the angle between them. So,
I was able to use the cosine law to determine y.
In the right triangle that contained u, I knew the
opposite side, 750.0 m, and the hypotenuse, y.
With these values, I was able to determine the
value of sin u.
I used the inverse sine to calculate u.
The angle of depression from the balloon
to Exit 83 is about 23°.
I rounded my answer to the nearest degree.
Reflecting
A.
Why do you think Vlad started by using the right triangle that
contained x instead of the right triangle that contained y?
B.
Vlad used the cosine law to determine y. Could he have used another
strategy to determine y? Explain.
C.
Could Vlad have calculated the value of u using the sine law? Explain.
APPLY the Math
EXAMPLE 2
Solving a problem that involves directions
The captain of a boat leaves a marina and heads due west for 25 km.
Then the captain adjusts the course of his boat and heads N30°E
for 20 km. How far is the boat from the marina?
Audrey’s Solution
N
Q
20 km
E
r
30°
60°
R
NEL
25 km
S
I drew a diagram to represent this
situation. The directions north
and west are perpendicular to
each other. Since N30°E means
that the boat travels along a line
30° east of north, I was able to
determine ∠QRS by subtracting
30° from 90°.
Communication
Tip
Directions are often stated in
terms of north and south on a
compass. For example, N30°E
means travelling in a direction
30° east of north. S45°W
means travelling in a direction
45° west of south.
N N30°E
30°
W
E
45°
S45°W
S
Chapter 8
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r 2 = s 2 + q 2 - 2(s)(q)cos R
r 2 = 202 + 252 - 2(20)(25)cos 60°
r 2 = 525
To determine r, I used
the cosine law.
r = 2525
#
r = 22.9
The boat is about 23 km from the marina.
EXAMPLE 3
Solving a problem using acute and right triangles
A weather balloon is directly between two tracking stations. The angles of
elevation from the two tracking stations are 55° and 68°. If the tracking
stations are 20 km apart, determine the altitude of the weather balloon.
Marnie’s Solution
C
I drew a diagram. I did not have enough information
about ^ ADC to calculate h, but I did have enough
information about ^ ACB to calculate x. I reasoned
that if I could determine the hypotenuse, x, in the
right triangle that contained the 55° angle, then I
could use a primary trigonometric ratio to calculate
the altitude, h, of the balloon.
x
h
55°
A
68°
D
20 km
B
∠ACB = 180° - 55° - 68°
= 57°
20
x
=
sin 68°
sin 57°
20
b
x = sin 68° a
sin 57°
#
x = 22.1
sin A =
h
x
h
sin 55° =
22.1
(22.1)(sin 55°) = h
#
18.1 = h
I knew that the angles in ^ ACB added up to 180°.
I knew ∠ACB and the side opposite it in ^ ACB.
I also knew ∠B, which is the angle opposite side x.
I used the sine law to write a proportion that
related these values. Then I solved for x.
^ ACD is a right triangle in which h is opposite
the 55° angle and x is the hypotenuse.
I used the sine ratio to write an equation. Then
I solved for h.
The altitude of the weather balloon is about 18 km.
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8.5
In Summary
Key Ideas
• If a real-world problem can be modelled using an acute triangle,
the sine law or cosine law, sometimes along with the primary
trigonometric ratios, can be used to determine unknown measurements.
• Drawing a clearly labelled diagram makes it easier to select a strategy
for solving the problem.
Need to Know
• To decide whether you need to use the sine law or the cosine law,
consider the information given about the triangle and the measurement
to be determined.
Information Given
Measurement
To Be
Determined
Use
two sides and the angle opposite one of the sides
angle
sine
law
two angles and a side
side
sine
law
two sides and the contained angle
side
cosine
law
three sides
angle
cosine
law
CHECK Your Understanding
1. Explain how you would determine the measurement of the indicated
angle or side in each triangle.
A
a)
b)
␪
c)
A
A
␪
2.7 cm
c
3.0 cm
2.9 cm
60°
B
NEL
3.1 cm
C
B
47°
1.8 cm C
B 2.0 cm C
Chapter 8
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2. Use the strategies you described to determine the measurements
of the indicated angles and sides in question 1.
PRACTISING
3. The angle between two equal sides of an isosceles triangle is 52°.
K
4. A boat leaves Oakville and heads due east for 5.0 km as shown in the
N
O
60°
4.0 km
diagram at the left. At the same time, a second boat travels in a direction
S60°E from Oakville for 4.0 km. How far apart are the boats when they
reach their respective destinations?
E
5.0 km
30°
Each of the equal sides is 18 cm long.
a) Determine the measures of the two equal angles in the triangle.
b) Determine the length of the third side.
c) Determine the perimeter of the triangle.
?
5. A radar operator on a ship discovers a large sunken vessel lying flat on
the ocean floor, 200 m directly below the ship. The radar operator
measures the angles of depression to the front and back of the sunken
ship to be 56° and 62°. How long is the sunken ship?
6. The base of a roof is 12.8 m wide as shown in the diagram at the left.
The rafters form angles of 48° and 44° with the horizontal. How long,
to the nearest tenth of a metre, is each rafter?
48°
44°
7. A flagpole stands on top of a building that is 27 m high. From a point on
12.8 m
the ground some distance away, the angle of elevation to the top of the
flagpole is 43°. The angle of elevation to the bottom of the flagpole is 32°.
a) How far is the point on the ground from the base of the building?
b) How tall is the flagpole?
8. Two ships, the Albacore and the Bonito, are 50 km apart. The Albacore
A
is N45°W of the Bonito. The Albacore sights a distress flare at S5°E.
The Bonito sights the distress flare at S50°W. How far is each ship
from the distress flare?
9. Fred and Agnes are 520 m apart. As Brendan flies overhead
in an airplane, they measure the angle of elevation of the airplane.
Fred measures the angle of elevation to be 63°. Agnes measures it
to be 36°. What is the altitude of the airplane?
Career Connection
Pilots and flight engineers
transport people, goods,
and cargo. Some test aircraft,
monitor air traffic, rescue
people, or spread seeds
for reforesting.
450
10. The Nautilus is sailing due east toward a buoy. At the same time, the
Porpoise is approaching the buoy heading N42°E. If the Nautilus is
5.4 km from the buoy and the Porpoise is 4.0 km from the Nautilus, on
a heading of S46°E, how far is the Porpoise from the buoy?
8.5 Solving Acute Triangle Problems
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8.5
11. Two support wires are fastened to the top of a satellite dish tower from
points A and B on the ground, on either side of the tower. One wire
is 18 m long, and the other wire is 12 m long. The angle of elevation
of the longer wire to the top of the tower is 38°.
a) How tall is the satellite dish tower?
b) How far apart are points A and B?
12. A regular pentagon is inscribed in a circle with radius 10 cm as shown
T
in the diagram at the right. Determine the perimeter of the pentagon.
13. Ryan is in a police helicopter 400 m directly above a highway. When
10 cm
he looks west, the angle of depression to a car accident is 65°. When
he looks east, the angle of depression to the approaching ambulance
is 30°.
a) How far away is the ambulance from the scene of the accident?
b) The ambulance is travelling at 80 km/h. How long will it take
the ambulance to reach the scene of the accident?
O
14. The radar screen in an air-traffic control tower shows that two
airplanes are at the same altitude. According to the range finder, one
airplane is 100 km away, in the direction N60°E. The other airplane
is 160 km away, in the direction S50°E.
a) How far apart are the airplanes?
b) If the airplanes are approaching the airport at the same speed,
which airplane will arrive first?
15. In a parallelogram, two adjacent sides measure 10 cm and 12 cm.
The shorter diagonal is 15 cm. Determine, to the nearest degree,
the measures of all four angles in the parallelogram.
16. Create a real-life problem that can be modelled by an acute triangle.
C
Then describe the problem, sketch the situation in your problem, and
explain what must be done to solve it.
Extending
17. From the top of a bridge that is 50 m high, two boats can be seen
anchored in a marina. One boat is anchored in the direction S20°W,
and its angle of depression is 40°. The other boat is anchored in the
direction S60°E, and its angle of depression is 30°. Determine the
distance between the two boats.
18. Two paper strips, each 5 cm wide, are laid across each other at an angle
30°
of 30°, as shown at the right. Determine the area of the overlapping
region. Round your answer to the nearest tenth of a square centimetre.
NEL
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Chapter Review
FREQUENTLY ASKED Questions
Study
Aid
Q:
To use the cosine law, what do you need to know about
a triangle?
A:
You need to know the measurements of three sides, or two sides and
the contained angle in the triangle. You can calculate the length of
a side if you know the measure of the angle that is opposite the side,
as well as the lengths of the other two sides. You can calculate the
measure of an angle if you know the lengths of all three sides.
• See Lesson 8.3 and
Lesson 8.4, Examples 1 to 3.
• Try Chapter Review
Questions 8 to 10.
EXAMPLE
Can you use the cosine law to determine the length of RQ? Explain.
8 cm
P
72°
6 cm
R
Solution
Yes. You know the lengths of two sides and
the angle that is opposite the side you want
to determine. Solving for p will give you
the measure of RQ.
p 2 = 82 + 62 - 2(8)(6)cos 72°
Q
Study
Aid
Q:
When solving a problem that can be modelled by an acute
triangle, how do you decide whether to use the primary
trigonometric ratios, the sine law, or the cosine law?
A:
Draw a clearly labelled diagram of the situation to see what you know.
• If the diagram involves one or more right triangles, you might be
able to use a primary trigonometric ratio.
• Use the sine law if you know the lengths of two sides and the
measure of one opposite angle, or the measures of two angles and
the length of one opposite side.
• Use the cosine law if you know the lengths of all three sides,
or two sides and the angle between them.
You may need to use more than one strategy to solve some problems.
• See Lesson 8.5,
Examples 1 to 3.
• Try Chapter Review
Questions 11 to 13.
452
Chapter Review
NEL
Chapter Review
PRACTICE Questions
Lesson 8.1
Lesson 8.4
1. Jane claims that she can draw an acute triangle
using the following information: a ⫽ 6 cm,
b ⫽ 8 cm, c ⫽ 10 cm, ∠ A ⫽ 30°, and
∠ B ⫽ 60°. Is she correct? Explain.
8. Calculate the indicated side length or angle
measure.
a)
87°
2. Which of the following are not correct for acute
triangle DEF ?
f
d
a)
=
sin D
sin F
sin D
sin E
b)
=
e
d
␪
x
15.0 m
14.0 m
6.0 m
d)
sin F
d
=
sin D
f
3. Calculate the indicated side length or angle
measure in each triangle.
a)
b)
67°
x
35.4 cm
58°
51°
31.2 cm
␪
4. In ^ ABC, ∠ B ⫽ 31°, b ⫽ 22 cm, and
c ⫽ 12 cm. Determine ∠ C.
5. Solve ^ ABC, if ∠ A ⫽ 75°, ∠ B ⫽ 50°, and
9. Solve ^ ABC, if ∠ A ⫽ 58°, b ⫽ 10.0 cm, and
c ⫽ 14.0 cm.
10. Two airplanes leave an airport at the same time.
One airplane travels at 355 km/h. The other
airplane travels at 450 km/h. About 2 h later,
they are 800 km apart. Determine the angle
between their paths.
Lesson 8.5
11. From the top of an 8 m house, the angle of
elevation to the top of a flagpole across the street
is 9°. The angle of depression is 22° to the base
of the flagpole. How tall is the flagpole?
9°
the side between these angles is 8.0 cm.
6. Allison is flying a kite. She has released the
entire 150 m ball of kite string. She notices that
the string forms a 70° angle with the ground.
Marc is on the other
side of the kite and
sights the kite at an
150 m
angle of elevation of
70°
30°
30°. How far is Marc
A
M
from Allison?
Lesson 8.3
7. Which of these is not a form of the cosine law
for ^ ABC ? Why not?
a) a 2 = b 2 + c 2 - 2 bc cos B
b) c 2 = a 2 + b 2 - 2 ab cos C
c) b 2 = a 2 + c 2 - 2 ac cos B
NEL
7.0 m
c) f sin E = e sin F
Lesson 8.2
24.5 m
b)
5.0 m
22°
8m
12. A bush pilot delivers supplies to a remote camp
by flying 255 km in the direction N52°E. While
at the camp, the pilot receives a radio message to
pick up a passenger at a village. The village is
85 km S21°E from the camp. What is the total
distance that the pilot will have flown by the
time he returns to his starting point?
13. A canoeist starts from a dock and paddles
2.8 km N34°E. Then she paddles 5.2 km N65°W.
What distance, and in which direction, should a
second canoeist paddle to reach the same location
directly, starting from the same dock?
Chapter 8
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8
Process
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Chapter Self-Test
Checklist
✔ Questions 1, 5, and 7:
1. Determine the indicated side length or angle measure in each triangle.
a)
b)
A
A
Did you use the given
representations to help
you select an appropriate
strategy?
on the information needed
to use the sine law as you
sketched and labelled the
triangle?
✔ Questions 4, 6, and 8: Did
you make connections
between each situation and
acute triangle trigonometry?
✔ Question 9: Did you
c
C
␪
6.0 cm
78°
4.05 cm
69°
4.25 cm
B
C
B
2. In ^ PQR, ∠ P = 80°, ∠ Q = 48°, and r = 20 cm. Solve ^ PQR.
3. a) Sketch an acute triangle. Label three pieces of information (side
lengths or angle measures) so that the sine law can be used to
determine at least one of the unknown side lengths or angle measures.
b) Use the sine law to determine one unknown side length or angle
measure in your triangle.
4. The radar screen of a Coast Guard rescue ship shows that two boats
communicate your
thinking clearly?
are in the area. According to the range finder, one boat is 70 km away,
in the direction N45°E. The other boat is 100 km away, in the
direction S50°E. How far apart are the two boats?
5. An engineer wants to build a bridge over a river from point A to point B
A
B
4.1 cm
82°
✔ Question 3: Did you reflect
72°
Page 454
as shown in the diagram at the left. The distance from point B to
point C is 515.0 m. The engineer uses a transit to determine that ∠ B
is 72° and ∠ C is 53°. Determine the length of the finished bridge.
6. A parallelogram has adjacent sides that are 11.0 cm and 15.0 cm long.
53°
515.0 m
C
The angle between these sides is 50°. Determine the length of the
shorter diagonal.
7. Terry is designing a new triangular patio.
A
The diagram at the right shows the
dimensions of the patio. Calculate the area
of the patio.
9.0 m
B
7.1 m
8.5 m
C
8. Points P and Q lie 240 m apart on opposite sides of a communications
tower. The angles of elevation to the top of the tower from P and Q
are 50° and 45°, respectively. Calculate the height of the tower.
9. In an acute triangle, two sides are 2.4 cm and 3.6 cm. One of the angles
is 37°. How can you determine the third side in the triangle? Explain.
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Chapter Self-Test
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Chapter Task
Dangerous Triangles
Some people claim that there is a region in the eastern end of Lake Ontario
near Kingston, called the Marysburgh Vortex, where more than two-thirds
of the shipwrecks in the lake are found. They say that the Marysburgh
Vortex is similar to the Bermuda Triangle because of its strange habit of
swallowing boats and airplanes. There are estimates of up to 450 wrecks in
the Marysburgh Vortex, with about 80 of them in the area from Kingston
to Prince Edward County. There are estimates of about 1000 wrecks in the
Bermuda Triangle.
Kingston
Bermuda
U.S.A
Marysburgh
Vortex
Prince
Edward
County
Miami
50°
ATLANTIC
OCEAN
Bermuda
Triangle
74°
1660 km
San Juan,
Puerto Rico
0
10
20 km
Mexico
?
Is the Marysburgh Vortex more dangerous than
the Bermuda Triangle?
A.
Use the map to estimate the lengths of the sides that form the triangle
of the Marysburgh Vortex.
B.
Use a strategy that does not involve trigonometry to determine
the area of the triangle for part A.
C.
Use trigonometry to calculate the area of the triangle for part A.
D.
Compare your answers for parts B and C.
E.
Calculate the area of the Bermuda Triangle.
F.
Which region do you think is more dangerous for sailing?
Justify your decision.
NEL
Task
Checklist
✔ Did you draw labelled
diagrams for the problem?
✔ Did you show your work?
✔ Did you provide appropriate
reasoning?
✔ Did you explain your
thinking clearly?
Chapter 8
455
Chapters
7–8
Cumulative Review
Multiple Choice
5. What is the value of x?
A. 3.00 cm
B. 6.80 cm
C. 9.70 cm
D. 5.00 cm
1. Which triangle is similar to ^ ABC ?
C
3.6 cm
79°
4.2 cm
5.0 cm
A. ^ DEF, with ∠ D = 45°, ∠ E = 62°
B. ^ DEF, with f = 12.5, d = 9.0 cm,
A.
B.
C.
D.
e = 10.5 cm
C. ^ DEF, with ∠ F = 79°, d = 7.2 cm,
e = 12.6 cm
D. ^ DEF, with f = 10 cm, d = 10.8 cm,
of an isosceles right triangle. He has created a
scale diagram. The lengths of the perpendicular
sides in the scale diagram are 7 cm, and the
hypotenuse of the real garden will be 3 m long.
What is the area of the real garden?
A. 450 m2
C. 1033 m2
2
B. 2.25 m
D. 4.5 m2
A
A. sin B =
c
b
4. What is the value of u to the nearest degree?
␪
7.1 cm
Cumulative Review
x
M
50°
C
12 cm
B
C
leaning against a
vertical wall. The foot
of the ladder is 2.0 m
from the base of the
wall. What is the
angle formed by the
ladder and the
ground?
A. 73.7°
B. 74.9°
T
8.0 m
F 2.0 m B
C. 75.5°
D. 76.6°
9. What is the area of this triangle?
C
6.33 cm
7.7 cm
456
9.3 cm
7.8 cm
3.9 cm
18.7 cm
8. An 8.0 m ladder is
3. Which statement about ^ ABC is true?
a
A
building, at a dump truck that is parked on the
road. The angle of depression to the front of the
truck is 58°, and the building is 37 m tall. What
is the distance between the base of the building
and the front of the truck?
A. 41 m
C. 59 m
B. 23 m
D. 27 m
2. Lucas is designing a flower garden in the shape
A. 25°
B. 65°
59.0°
7. Leo is looking down, from the roof of a
∠ E = 56°
B
x
6. If BM = MC and BC = 12 cm, what is
A
the value of x?
A
a
c
a
B. cos A =
c
a
C. tan B =
b
D. sin A = cos B
5.83 cm
B
45°
B
C
A
C. 23°
D. 67°
18.5°
9.00 cm
A. 28.5 cm2
B. 27.0 cm2
B
C. 2.0 cm2
D. 9.0 cm2
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Cumulative Review
10. Stephanie is 117 cm tall, and her eyes are
106 cm off the ground. She is using her
new binoculars to look at a bird that is perched
in a tree. The angle of elevation is 28°, and
Stephanie is 25.0 m from the base of the tree.
What is the height of the bird in the tree?
A. 13.3 m
C. 23.1 m
B. 12.8 m
D. 14.4 m
11. Greg and Kristen are on opposite ends of a zip
line that crosses a gorge. Greg went across the
gorge first, and he is now on a ledge that is 15 m
above the bottom of the gorge. Kristen is at the
top of a cliff that is 72 m above the bottom of
the gorge. Jon is on the ground at the bottom of
the gorge, below the zip line. He sees Kristen at
a 65° angle of elevation and Greg at a 35° angle
of elevation. What is the width of the gorge,
to the nearest metre?
A. 165 m
C. 55 m
B. 152 m
D. 106 m
12. In ^ ABC, ∠ A = 56°, ∠ B = 64°, and
c = 6.0 cm. What is the length of side a?
C. 0.5 cm
D. 0.96 cm
A. 5.7 cm
B. 6.3 cm
13. What is the measure of u?
27°
38.4 cm
A. 74°
B. 12°
C. 57°
D. 13°
14. In ^ ABC, ∠A = 57°, b = 5.0 cm, and
c = 8.0 cm. What is the length of side a?
C
b 5.0 cm
57°
A
c 8.0 cm
A. 45.4 cm
B. 6.7 cm
NEL
of u?
11.0 m
7.0 m
15.0 m
A. 13°
B. 43°
C. 73°
D. 47°
16. In ^ ABC, ∠ A = 58°, b = 10.0 cm, and
c = 14.0 cm. Solve ^ ABC.
∠ B = 90°, ∠ C = 32°, a = 9.8 cm
∠ B = 86°, ∠C = 36°, a = 17.2 cm
∠ B = 44°, ∠ C = 78°, a = 12.2 cm
∠ B = 78°, ∠C = 44°, a = 12.2 cm
A.
B.
C.
D.
17. Which statements are true for ^ ABC ?
b
a
=
and
sin A
sin B
a 2 = b 2 + c 2 - 2 bc cos A
sin B
a
=
B.
and
sin A
b
a 2 = b 2 + c 2 - 2 bc cos A
b
a
=
C.
and
sin A
sin B
a 2 = b 2 + c 2 + 2 bc cos A
b
a
=
D.
and
sin A
sin B
c 2 = a 2 + b 2 - 2 ab cos A
A.
18. An isosceles triangle has sides that measure
81.2 cm
15. What is the measure
B
4 cm, 10 cm, and 10 cm. What is the area
of the triangle?
A. 38 cm2
C. 25 cm2
2
B. 10 cm
D. 20 cm2
19. A cyclist travels 4.50 km directly south, and
then turns and travels 6.80 km in the direction
N60°W. What distance, and in what direction,
will the cyclist need to travel to get back to her
starting point?
A. 5.00 km, N59.5°E
B. 6.00 km, N79.5°E
C. 5.00 km, N49.5°E
D. 4.00 km, N29.5°E
C. 9.4 cm
D. 11.5 cm
Chapters 7–8
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Investigations
Running Cable
extra
supports
required
A hydro company needs to deliver
power to a new subdivision beside
a lake. The nearest power station
is on the other side of the lake.
The power station sits on top of a
vertical cliff, 23 m above the lake.
The angle of depression from the
power station to the subdivision is
14°. There are two options
available:
power station
39 m
34 m
23 m
15 m
33 m
61 m
23 m
subdivision
Option A: Run a cable directly down the cliff at a cost of $12/m and then
underwater at a cost of $33/m.
Option B: Run a cable along a series of towers around the lake. Each
tower would be 15 m tall. Running the cable this way would
cost $17/m once the cable has reached the first tower. To get
from the power station to the first tower, one extra support
would be required for every 5° change in elevation. Each extra
support would cost $25.
20. Determine which option is less costly. Explain how you know.
The Great Pyramid
The Great Pyramid of Giza in Egypt has a square base with sides that are
232.6 m in length. The distance from the top of the pyramid to each
corner of the base was originally 221.2 m.
E
221.2 m
C
D
G
F
A
232.6 m
B
21. a) Determine, to the nearest degree, the angle that each face makes
with the base (∠ EGF ).
b) Determine, to the nearest degree, the size of the apex angle
of a face of the pyramid (∠ AEB ).
458
Cumulative Review
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Appendix
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Review of Essential Skills
and Knowledge
A–1
Operations with Integers
460
A–2
Operations with Rational Numbers
462
A–3
Exponent Laws
463
A–4
The Pythagorean Theorem
464
A–5
Evaluating Algebraic Expressions and Formulas
466
A–6
Determining the Intercepts of Linear Relations
467
A–7
Graphing Linear Relations
469
A–8
Expanding and Simplifying Algebraic Expressions
471
A–9
Solving Linear Equations Algebraically
472
A–10
First Differences and Rate of Change
473
A–11
Creating Scatter Plots and Lines or Curves of Good Fit
475
A–12
Interpolating and Extrapolating
477
A–13
Transformations of Two-Dimensional Figures
479
A–14
Ratios, Rates, and Proportions
481
A–15
Properties of Triangles and Angle Relationships
483
A–16
Congruent Figures
485
Appendix A: Review of Essential Skills and Knowledge
Appendix A
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A–1 Operations with Integers
Integers are all the counting numbers, their opposites, and zero.
Set of integers: I {…, 3, 2, 1, 0, 1, 2, 3, …}
Addition
To add two integers,
• if the integers have the same sign, then the sum has the same sign:
12 (5) 17
• if the integers have different signs, then the sum takes the sign of the
integer farthest from 0: 18 (5) 13
Subtraction
To subtract one integer from another integer, add the opposite.
15 (8) 15 8
7
Multiplication and Division
To multiply or divide two integers,
• if the two integers have the same sign, then the answer is positive:
6 * 8 = 48, -36 , (-9) = 4
• if the two integers have different signs, then the answer is negative:
- 5 * 9 = - 45, 54 , (-6) = - 9
More Than One Operation
Follow the order of operations.
B
Brackets
E
Exponents
D Division
M Multiplication
f
from left to right
A
S
f
from left to right
Addition
Subtraction
EXAMPLE
a)
b)
c)
d)
-10 + (-12)
-12 - (-7)2
-11 + ( -4) + 12 + (-7) + 18
-6 * 9 , 3
20 + ( -12) , (-3)
e)
(-4 + 12) , (-2)
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Solution
Appendix A
a) - 10 + ( - 12) = - 22
b) - 12 - (- 7)2 = - 12 - 49
= - 12 + (-49)
= - 61
c) - 11 + (- 4) + 12 + ( -7) + 18
= - 22 + 30
= 8
d) - 6 * 9 , 3
= - 54 , 3
= - 18
20 + (- 12) , ( - 3)
e)
( - 4 + 12) , (- 2)
20 + 4
=
8 , ( - 2)
24
=
-4
= -6
Appendix B
Glossary
Practice
1. Evaluate.
4. Evaluate.
a) 6 (3)
b) 12 (13)
c) 17 7
d) 23 9 (4)
e) 24 36 (6)
f ) 32 (10) (12) 18 (14)
2. Which sign would make each statement true:
, , or ?
a) 5 4 3 3 ■ 4 3 1 (2)
b) 4 6 6 8 ■ 3 5 (7) 4
c) 8 6 (4) 5 ■ 5 13 7 (8)
d) 5 13 7 2 ■ 4 5 (3) 5
3. Evaluate.
NEL
5. Evaluate.
a)
b)
c)
d)
e)
f)
-12 - 3
-3 - 2
-18 + 6
(-3)( -4)
(-16 + 4) , 2
8 , (- 8) + 4
-5 + (-3)( - 6)
(-2)2 + ( -3)2
(-2)3(9)
62 , ( -4)
(5)2 - ( -3)2
[( -2)(4)]2
Appendix A: Review of Essential Skills and Knowledge
Index
a) 11 (5)
b) 3(5)(4)
c) 35 (5)
d) 72 (9)
e) 5(9) (3)(7)
f ) 56 [(8)(7)] 49
Answers
a) (-3)2 - ( -2)2
b) ( -5)2 - ( -7) + (-12)
c) -4 + 20 , (- 4)
d) -3(-4) + 82
e) -16 - [( -8) , 2]
f ) 8 , (-4) + 4 , (-2)2
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A–2 Operations with Rational Numbers
Rational numbers are numbers that can be expressed as the quotient
of two integers, where the divisor is not zero.
a
Set of rational numbers: Q = e ` a, b I, b Z 0 f
b
Addition and Subtraction
To add or subtract rational
numbers, determine a common
denominator.
Multiplication
To multiply rational numbers, first
reduce to lowest terms (if possible).
a
c
ac
* =
b
d
bd
Division
To divide by a rational number,
multiply by the reciprocal.
c
a
d
a
ad
, = * =
c
b
d
b
bc
EXAMPLE 1
Simplify.
More Than One Operation
Follow the order of operations.
EXAMPLE 2
3
3
-2
+
5
-2
10
Simplify.
-4
-3
3
*
,
4
5
7
Solution
Solution
3
3
-4
- 15
3
-2
+
=
+
5
-2
10
10
10
10
- 4 - 15 - 3
=
10
- 22
1
=
or - 2
10
5
3
-4
-3
3
-4
7
*
,
= *
*
4
5
7
4
5
-3
1
-1
3
-4
=
*
4
5
*
7
-3
1
=
-1
-7
2
or 1
-5
5
Practice
1. Evaluate.
1
-3
+
4
4
1
-2
b)
2
3
a)
c)
-1
2
*
3
-5
1
3
d) - 4 * a-7 b
6
4
-3
-2
,
3
8
1
1
f ) a- 2 b , a- 3 b
3
2
3
1
-2
c) a b a
ba
b
5
-6
3
-2 2 1 3
d) a
b a
b
3
-2
e) a
e)
2. Evaluate.
-2
-1
1
- a
+
b
5
10
-2
-1
-3 -3
b)
a
b
5
4
4
a)
462
Appendix A: Review of Essential Skills and Knowledge
-2
1
5
-1
+
b , a
b
5
-2
-8
2
-4
-3
1
-1
f)
* 5
5
3
5
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A–3 Exponent Laws
exponent
34
an
and
4 factors of 3
34 = (3)(3)(3)(3)
Operations with powers follow a set of rules.
Appendix A
base
34 and a n are called powers.
n factors of a
a n = (a)(a)(a)…(a)
Description
Algebraic Expression
Example
Multiplication
When the bases are the
same, keep the base the
same and add exponents.
(am)(an) = am + n
(54)(5-3) = 54 + (-3)
= 54 - 3
= 51
= 5
Division
When the bases are the same,
keep the base the same and
subtract exponents.
am
= am-n
an
46
= 46 - (-2)
4-2
= 46 + 2
= 48
Power of a Power
Keep the base, and multiply
the exponents.
(am)n = amn
(32)4 = 3(2)(4)
= 38
Appendix B
Rule
EXAMPLE
Simplify and evaluate.
3(37) , (33)2
Solution
Glossary
3(37) , (33)2 = 31 + 7 , 33 * 2
= 38 , 36
= 38 - 6
= 32
= 9
Practice
1. Evaluate to three decimal places, if necessary.
b) 50
c)
32
e)
(-5)3
1 3
f) a b
2
d) - 32
2. Evaluate.
a) 30 + 50
b) 22 + 33
c) 52 - 42
3
98
97
21552
b)
53
NEL
c) (45)(42)3
d)
a) (x 5)(x 3)
b) (m 2)(m 4)(m 3)
c) ( y 5)( y 2)
d) (a b)c
1x 521x 32
e)
x2
x4 3
f) a 3 b
y
5. Simplify.
a) (x 2y 4)(x 3y 2)
b) (2m3)2(3m2)3
15x 222
c)
15x 220
d) (4u3v2)2 (2u2v3)2
Index
1
2
d) a b a b
2
3
5
e) -2 + 24
1 2
1 2
f) a b + a b
2
3
3. Evaluate to an exact answer.
a)
2
4. Simplify.
Answers
a)
42
13221332
13422
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A–4 The Pythagorean Theorem
a
right angle
c
hypotenuse
b
The three sides of a right triangle are related to each other in a unique
way, through an important mathematical relationship called the
Pythagorean theorem.
Every right triangle has a side called the hypotenuse, which is always the
longest side and opposite the right angle. According to the Pythagorean
theorem, the area of the square of the hypotenuse is equal to the sum
of the areas of the squares of the other two sides.
c2
a2
b2
a2
⫹
b2
⫽
c2
EXAMPLE
Determine each indicated side length. Round your answers
to one decimal place, if necessary.
a)
12 cm
5 cm
?
b)
14.0 cm
20.0 cm
?
Solution
a) Let a = 5, b = 12, and c = ?.
a2 + b2 = c 2
52 + 122 = c 2
25 + 144 = c 2
169 = c 2
2169 = c
13 = c
The length is 13 cm.
b) Let a = 14.0, b = ?, and c = 20.0.
a2 + b2 = c 2
14.02 + b 2 = 20.02
196.00 + b 2 = 400.00
b 2 = 400.00 - 196.00
b 2 = 204.00
b = 2204.00
b 14.3
The length is about 14.3 cm.
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Practice
4. Determine the length of the diagonals
right triangle.
a)
Appendix A
1. Write the Pythagorean theorem for each
in each rectangle to one decimal place.
a)
5m
x
6 cm
10 m
8 cm
b)
Appendix B
b)
6 cm
c
13 cm
3 cm
c)
5.2 cm
6 cm
Glossary
c)
5.2 cm
d) 1.2 m
9m
4.8 m
y
5. An isosceles triangle has a hypotenuse that is
15.0 cm long. Determine the length of the two
equal sides.
5m
a
d)
of the shadow is 100.0 m from the top of the
building and 72.0 m from the base of the
building. How tall is the building?
8.5 cm
Answers
3.2 cm
6. An apartment building casts a shadow. The tip
2. Calculate the length of the unknown side
in each triangle in question 1. Round your
answers to one decimal place, if necessary.
100.0 m
3. Determine the value of each unknown measure
NEL
72.0 m
Index
to two decimal places, if necessary.
a) a 2 = 52 + 132
b) 102 = 82 + m 2
c) 262 = b 2 + 122
d) 2.32 + 4.72 = c 2
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A–5 Evaluating Algebraic Expressions
and Formulas
To evaluate algebraic expressions and formulas, substitute the given numbers
for the variables. Then follow the order of operations to calculate the answer.
EXAMPLE 1
EXAMPLE 2
Determine the value of 2x 2 - y
if x = - 2 and y = 3.
The formula for calculating the
volume of a cylinder is V = pr 2h.
Determine the volume of a cylinder
with a radius of 2.5 cm and a height
of 7.5 cm.
Solution
Solution
2x 2 - y = 21-222 - 3
V = pr 2h
13.14212.52217.52
= 13.14216.25217.52
147.2
= 2142 - 3
= 8 - 3
= 5
The volume is about 147.2 cm3.
Practice
1. Determine the value of each expression
3. a) The formula for the area of a triangle is
if x = - 5 and y = - 4.
a) - 4x - 2y
b) - 3x - 2y 2
c) (3x - 4y)2
y
x
d)
y
x
1
2
and y = , calculate the value
2
3
of each expression.
a) x + y
b) x + 2y
c) 3x - 2y
1
1
d)
x - y
2
2
2. If x = -
1
2
A bh. Determine the area of a triangle
if b 13.5 cm and h 12.2 cm.
b) The area of a circle is calculated using the
formula A = pr 2. Determine the area
of a circle with a radius of 4.3 m.
c) The hypotenuse of a right triangle, c, is
calculated using the formula
c = 2a 2 + b 2. Determine the length of
the hypotenuse if a = 6 m and b = 8 m.
d) The volume of a sphere is calculated using
4
the formula V = pr 3. Determine the
3
volume of a sphere with a radius of 10.5 cm.
10.5 cm
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Appendix A
A–6 Determining the Intercepts
of Linear Relations
A linear relation of the general form Ax + By + C = 0 has an x-intercept
and a y-intercept. These are the points where the line Ax + By + C = 0
crosses the x-axis and the y-axis.
EXAMPLE 1
AX BY C 0 FORM
Determine the x- and y-intercepts of the linear relation 2x + y - 6 = 0.
Appendix B
Solution
The x-intercept is where the relation crosses the x-axis. The equation
of the x-axis is y = 0, so substitute y = 0 into 2x + y - 6 = 0.
2x + 0 - 6 = 0
2x = 6
x = 3
To determine the y-intercept, substitute x = 0 into 2x + y - 6 = 0.
2102 + y - 6 = 0
y = 6
Glossary
The x-intercept is at (3, 0), and the y-intercept is at (0, 6).
EXAMPLE 2
ANY FORM
Determine the x- and y-intercepts of the linear relation 3y = 18 - 2x.
Solution
Answers
To determine the x-intercept, substitute y = 0 into the equation.
3y = 18 - 2x
3102 = 18 - 2x
2x = 18
x = 9
To determine the y-intercept, substitute x = 0 into the equation.
3y = 18 - 2x
3y = 18 - 2102
3y = 18
y = 6
NEL
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Index
The x-intercept is at (9, 0), and the y-intercept is at (0, 6).
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A special case is when the linear relation is a horizontal or vertical line.
EXAMPLE 3
Determine either the x- or the y-intercept of each linear relation.
a) 2x = - 14
b) 3y + 48 = 0
Solution
a) 2x = - 14 is a vertical line, so it has no y-intercept.
To determine the x-intercept, solve for x.
2x = - 14
x = -7
The x-intercept is located at (7, 0).
b) 3y + 48 = 0 is a horizontal line, so it has no x-intercept.
To determine the y-intercept, solve for y.
3y + 48 = 0
3y = - 48
y = - 16
The y-intercept is located at (0, 16).
Practice
1. Determine the x- and y-intercepts of each
3. Determine either the x-intercept or y-intercept
linear relation.
a) x + 3y - 3 = 0
b) 2x - y + 14 = 0
c) - x + 2y + 6 = 0
d) 5x + 3y - 15 = 0
e) 10x - 10y + 100 = 0
f ) - 2x + 5y - 15 = 0
2. Determine the x- and y-intercepts of each
of each linear relation.
a) x = 13
b) y = - 6
c) 2x = 14
d) 3x + 30 = 0
e) 4y = - 6
f ) 24 - 3y = 0
4. A ladder resting against a wall is modelled by
linear relation.
a) x = y + 7
b) 3y = 2x + 6
c) y = 4x + 12
d) 3x = 5y - 30
e) 2y - x = 7
f ) 12 = 6x - 5y
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the linear relation 2y + 9x = 13.5. The x-axis
represents the ground, and the y-axis represents
the wall.
a) Determine the intercepts of the relation.
b) Use the intercepts to graph the relation.
c) What does this tell you about the foot
of the ladder and the top of the ladder?
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A–7 Graphing Linear Relations
Appendix A
The graph of a linear relation (Ax + By + C = 0) is a straight line. The
graph can be drawn if at least two ordered pairs of the relation are known.
This information can be determined in several different ways.
EXAMPLE 1
TABLE OF VALUES
Sketch the graph of 2y = 4x - 2.
Solution
Choose values for x, and substitute these
values into the equation to calculate y.
y
1
2( -1) - 1 = - 3
0
2(0) - 1 = - 1
1
2(1) - 1 = 1
2
2(2) - 1 = 3
EXAMPLE 2
Plot the ordered pairs
as points, and join them
with a straight line.
5
0
⫺5
y
5x
2y ⫽ 4x ⫺ 2
Glossary
x
Appendix B
A table of values can be created. Express
the equation in the form y = mx + b.
2y
4x - 2
=
2
2
y = 2x - 1
⫺5
USING INTERCEPTS
Answers
Sketch the graph of 2x + 4y = 8.
Solution
The intercepts of the line can be calculated.
For the x-intercept, let y = 0.
2x + 4102 = 8
2x = 8
x = 4
The x-intercept is at (4, 0).
NEL
y
5
2x ⫹ 4y ⫽ 8
0
5
Index
For the y-intercept, let x = 0.
2102 + 4y = 8
4y = 8
y = 2
The y-intercept is at (0, 2).
Plot the ordered pairs
as points, and join them
with a straight line.
x
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EXAMPLE 3
USING THE SLOPE AND Y-INTERCEPT
Sketch the graph of y = 3x + 4.
Solution
When the equation is in the form
y = mx + b, m is the slope and
b is the y-intercept.
The line y = 3x + 4 has a slope
of 3 and a y-intercept of 4.
Plot the y-intercept, and use the
rise and the run of the slope to
locate another point on the line.
Join the points with a straight line.
y
7
6
rise ⫽ 3
5
4
3 run ⫽ 1
2 y ⫽ 3x ⫹ 4
x
1
0
⫺1
1 2 3
Practice
1. Express each equation in the form y = mx + b.
4. Graph each equation by determining
the intercepts.
a) x + y = 4
b) x - y = 3
c) 2x + 5y = 10
d) 3x - 4y = 12
a) 3y = 6x + 9
b) 2x - 4y = 8
c) 3x + 6y - 12 = 0
d) 5x = y - 9
2. Graph each equation using a table of values.
a) y = 3x - 1
1
b) y = x + 4
2
c) 2x + 3y = 6
d) y = 4
5. Graph each equation using the slope and
y-intercept.
a) y = 2x + 3
2
b) y = x - 1
3
3
c) y = - x - 2
4
d) 2y = x + 6
3. Determine the x- and y-intercepts of
each equation.
a) x + y = 10
b) 2x + 4y = 16
c) 50 - 10x - y = 0
y
x
d)
+ = 1
2
4
470
6. Graph each equation. Use any strategy.
Appendix A: Review of Essential Skills and Knowledge
a) y = 5x + 2
b) 3x - y = 6
2
c) y = - x + 4
3
d) 4x = 20 - 5y
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A–8 Expanding and Simplifying Algebraic Expressions
Appendix A
An algebraic expression may contain both numbers and letters. The letters are called
variables. An algebraic expression can only be simplified if it contains like terms. Like terms
must have the same variables and the same exponents. For example, 3x 2 and -5x 2 are like
terms and 2xy and 7yz are unlike terms.
A term is the product of a coefficient and a variable. In an algebraic expression, the terms
are separated by addition or subtraction signs. For example, 5x 2 - 3x + 7 has three terms,
where 5 and 3 are the coefficients, 7 is the constant, and x is the variable.
Description
Example
Collecting like terms
2a 3a 5a
Add or subtract the coefficients of the
terms that have the same variables
and exponents.
3a 2b 5a b
3a 5a 2b b
2a b
Distributive property
a(b c) ab ac
Multiply each term of the
binomial by the monomial.
Appendix B
An algebraic expression with one or more terms is called a polynomial. Simple
polynomials have special names: monomial (one term), binomial (two terms),
or trinomial (three terms).
Glossary
4a(2a 3b)
8a2 12ab
Practice
1. Identify the variable and the coefficient
in each expression.
c) 7c 4
b) - 13a
d) - 1.35m
4
y
7
5x
f)
8
e)
2. In each group of terms, which terms are
3. Identify each polynomial as a monomial,
a binomial, or a trinomial.
a) 6x 3 - 5x
d) -yxz 3
3
b) 5x y
e) 5x - 2y
2
c) 7 + 3x - 4x
f ) 3a + 5c - 4b
NEL
5. Expand.
a) 3(2x + 5y - 2)
b) 5x(x 2 - x + y)
c) m 2(3m 2 - 2n)
d) x 5y 3(4x 2y 4 - 2xy 5)
6. Expand and simplify.
a) 3x(x + 2) + 5x(x - 2)
b) -7h(2h + 5) - 4h(5h - 3)
c) 2m 2n(m 3 - n) - 5m 2n(3m 3 + 4n)
d) 3xy3(5x 2y 1) 2xy3(3y 2 7x)
Appendix A: Review of Essential Skills and Knowledge
471
Index
like terms?
a) a, 5x, - 3a, 12a, - 9x
b) c 2, 6c, - c, 13c 2, 1.25c
c) 3xy, 5x 2y, - 3xy, 9x 2y, 12x 2y
d) x 2, y 2, 2xy, - y 2, - x 2, – 4xy
a) 3x + 2y - 5x - 7y
b) 5x 2 - 4x 3 + 6x 2
c) (4x - 5y) - (6x + 3y) - (7x + 2y)
d) m 2n + p - (2p + 3m 2n)
Answers
a) 5x 3
4. Simplify.
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A–9 Solving Linear Equations Algebraically
Any mathematical sentence stating that two quantities are equal is called
an equation; for example, 5x + 6 = 2(x + 5) + 5.
A solution to an equation is a number that makes the left side equal to the
right side. The solution to 5x + 6 = 2(x + 5) + 5 is x = 3. When x is
replaced with 3, both sides of the equation result in 21.
To solve a linear equation, use inverse operations. When necessary,
eliminate fractions by multiplying each term in the equation by the lowest
common denominator. Eliminate brackets by using the distributive
property. Then isolate the variable. A linear equation has only one solution.
EXAMPLE 1
EXAMPLE 2
Solve.
- 3(x + 2) - 3x = 4(2 - 5x)
Solve.
Solution
Solution
- 31x + 22 - 3x = 412 - 5x2
- 3x - 6 - 3x = 8 - 20x
- 3x - 3x + 20x = 8 + 6
14x = 14
14
x =
14
x = 1
y - 2
y - 7
=
3
4
y - 2
y - 7
b = 12a
b
12 a
3
4
41y - 72 = 31y - 22
4y - 28 = 3y - 6
4y - 3y = - 6 + 28
y = 22
y - 2
y - 7
=
3
4
Practice
1. Solve.
3. Determine the solution to each equation.
a) 6x - 8 = 4x + 10
b) 2x + 7.8 = 9.4
c) 13 = 5m - 2
d) 13.5 - 2m = 5m + 41.5
e) 8(y - 1) = 4(y + 4)
f ) 4(5 - r) = 3(2r - 1)
x
= 20
5
2
b)
x = 8
5
3
c) 4 = m + 3
2
a)
5
y = 3 + 12
7
2
1
e) 3y - =
2
3
m
m
f) 4 = 5 +
3
2
d)
2. a) A triangle has an area of 15 cm2 and a base
4. A total of 209 tickets for a concert were sold.
of 5 cm. What is the height of the triangle?
b) A rectangular lot has a perimeter of 58 m
and is 13 m wide. How long is the lot?
There were 23 more student tickets sold than
twice the number of adult tickets. How many
of each type of ticket were sold?
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A–10 First Differences and Rate of Change
Appendix A
When the values of the independent variable x increase by the same
amount throughout a table of values, the differences between the successive
values of the dependent variable y form a table of first differences called a
difference table.
Appendix B
A difference table represents a linear relation if
• the first differences are the same in every row
• a single straight line can be drawn through all the points when they are
plotted on a grid
• the ratio of the difference between the values of x to the corresponding
difference between the values of y is the same for all pairs of points
in the table
Glossary
This ratio represents the rate of change between the variables and
is equivalent to the slope of the line.
¢y
y2 - y1
Rate of change slope =
x2 - x1
¢x
A table of values represents a nonlinear relation if
• the first differences vary; that is, they are not the same
for every row in the difference table
• a single smooth curve, not a straight line, can be drawn through
the points when they are plotted on a grid
EXAMPLE
This table shows the cost to rent a cement mixer for different lengths of time.
0
1
2
3
4
Cost ($)
45
60
75
90
105
Answers
Time (h)
a) Is the relation linear or nonlinear?
b) Graph the data.
c) Calculate the slope. What does the slope represent in this situation?
Solution
Cost ($)
0
45
1
60
2
75
3
90
4
105
First Difference
60 45 15
75 60 15
90 75 15
105 90 15
120
100
80
60
40
20
The first differences are constant, so the relation is linear.
NEL
140
b)
Index
Time (h)
Cost ($)
a)
Cost to Rent a Cement Mixer
y
0
x
1
2
3 4
Time (h)
5
6
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65 - 40
1 - 0
15
=
1
= 15
c) Slope =
The slope represents the rate of change. In this situation, the rate
of change is the rate at which the cost changes as time changes.
The cost increases by $15 for each additional hour.
Practice
1. For each set of data,
i) create a first differences table
ii) determine whether the relation is linear or nonlinear
iii) graph the relationship
a) speed of a falling ball
Time from Release (s)
0
1
2
3
4
5
6
7
8
Speed (m/s)
0.0
9.8
19.6
29.4
39.2
49.0
58.8
68.6
78.4
b) volumes of various cones with a height of 1 cm
Radius of Base (cm)
1
2
3
4
5
6
7
8
Volume of Cone (cm3)
1.047
4.189
9.425
16.755
26.180
37.699
51.313
67.021
c) cost of renting a car for a day
Distance Driven (km)
Cost ($)
0
10
20
30
40
50
60
70
80
45.00
46.50
48.00
49.50
51.00
52.50
54.00
55.50
57.00
1
2
3
4
5
6
7
5800
4850
3900
2950
2000
1050
100
d) value of a photocopier after several years
Age (years)
0
Value of Photocopier ($) 6750
e) population of a town
Year
2001
2002
2003
2004
2005
2006
2007
2008
Population
1560
1716
1888
2077
2285
2514
2765
3042
2. Determine the rate of change for each linear relation in question 1.
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Appendix A
A–11 Creating Scatter Plots and Lines
or Curves of Good Fit
A scatter plot is a graph that shows the relationship between two sets of
numeric data. The points in a scatter plot often show a general pattern, or
trend. A line that approximates a trend for the data in a scatter plot is
called a line of best fit.
A line of best fit passes through as many points as possible, with the
remaining points grouped equally above and below the line.
Appendix B
Data that have a positive correlation have a pattern that slopes up and to
the right. Data that have a negative correlation have a pattern that slopes
down and to the right. If the points nearly form a line, then the correlation
is strong. If the points are dispersed but still form a linear pattern, then the
correlation is weak.
A curve that approximates, or is close to, the data is called
a curve of good fit.
EXAMPLE 1
Glossary
a) Make a scatter plot to display the data in the table. Describe the kind
of correlation that the scatter plot shows.
b) Draw the line of best fit.
Long-Term Trends in Average Number of Cigarettes Smoked per Day
by Smokers Aged 15–19
1981
1983
1985
1986
1989
1990
1991
1994
1995
1996
Number per Day
16.0
16.6
15.1
15.4
12.9
13.5
14.8
12.6
11.4
12.2
Answers
Year
b)
Average Number of Cigarettes
Smoked/Day, Ages 15–19
1981
1985
1989
Year
1993
1997
Average Number of Cigarettes
Smoked/Day, Ages 15–19
18
16
14
12
10
8
6
4
2
0
1981
1985
1989
Year
1993
Index
18
16
14
12
10
8
6
4
2
0
Number ofcigarettes smoked/day
a)
Number ofcigarettes smoked/day
Solution
1997
The scatter plot shows a negative correlation.
NEL
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EXAMPLE 2
A professional golfer is taking part in a scientific investigation. Each time
she drives the ball from the tee, a motion sensor records the initial speed
of the ball. The final horizontal distance of the ball from the tee is also
recorded. Here are the results:
Speed (m/s)
10
16
19
22
38
Distance (m)
10
25
47
43
142 182 244 280
Solution
The scatter plot shows that a line of best
fit does not fit the data as well as an
upward-sloping curve does. Therefore,
sketch a curve of good fit.
50
54
Horizontal Distance of a Golf Ball
300
250
200
150
100
50
Distance (m)
Draw the line or curve of good fit.
43
0
10 20 30 40 50 60
Speed (m/s)
Practice
1. For each set of data,
i) create a scatter plot and draw the line of best fit
ii) describe the type of correlation the trend in the data displays
a) population of the Hamilton–Wentworth, Ontario, region
Year
1966
1976
1986
1996
1998
Population
449 116
529 371
557 029
624 360
618 658
b) percent of Canadians with less than Grade 9 education
Year
1976
1981
1986 1991 1996
Percent of the Population
25.4
20.7
17.7
14.3 12.4
2. In an experiment for a physics project, a marble is rolled up a ramp. A motion sensor detects
the speed of the marble at the start of the ramp, and the final height of the marble is recorded.
The motion sensor, however, may not be measuring accurately. Here are the data:
Speed (m/s)
1.2
2.1
2.8
3.3
4.0
4.5
5.1
5.6
Final Height (m) 0.07
0.21
0.38
0.49
0.86
1.02
1.36
1.51
a) Draw a curve of good fit for the data.
b) How accurate are the motion sensor’s measurements? Explain.
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A–12 Interpolating and Extrapolating
Appendix A
A graph can be used to make predictions about values that are not recorded and plotted. When a prediction
involves a point within the range of values, it is called interpolating. When the value of the independent
variable falls outside the range of recorded data, it is called extrapolating. Estimates from a scatter plot
are more reliable if the data show a strong positive or negative correlation.
EXAMPLE
Solution
Winning Times of Men’s 100 m Run
Name (Country)
Williams (Canada)
Time (s)
10.8
1932
Tolan (U.S.)
10.3
12
1936
Owens (U.S.)
10.3
10
1948
Dillard (U.S.)
10.3
1952
Remigino (U.S.)
10.4
6
1956
Morrow (U.S.)
10.5
4
0
1960
Hary (Germany)
10.2
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
a) Draw a scatter plot, and determine the line of best fit.
Year
1928
1964
Hayes (U.S.)
10.0
Year
1968
Hines (U.S.)
9.95
1972
Borzov (U.S.S.R.)
10.14
Locate 1940 on the x-axis. Follow the vertical line for
1940 up until it meets the line of best fit at about
10.5 s. For 1944, a reasonable estimate is about 10.4 s.
1976
Crawford (Trinidad)
10.06
1980
Wells (Great Britain)
10.25
1984
Lewis (U.S.)
9.99
1988
Lewis (U.S.)
9.92
12
1992
Christie (Great Britain)
9.96
10
1996
Bailey (Canada)
9.84
Time (s)
100 m Run
y
8
Glossary
x
Answers
b) Extend the x-axis to 2024. Then extend the line
of best fit to the vertical line through 2024.
y
8
6
x
Year
The vertical line for 2024 crosses the line of best fit at about 9.7 s. It would be difficult to predict much
farther into the future, since the winning times cannot continue to decline indefinitely.
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477
Index
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
2012
2016
2020
2024
Time (s)
100 m Run
4
0
NEL
Appendix B
The Summer Olympics were cancelled in 1940 and 1944 because of World War II.
a) Estimate what the men’s 100 m run winning times might have been in these years.
b) Predict what the winning time might be in 2024.
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Practice
1. This scatter plot shows the gold medal throws
3. Explain why values you obtained by
in the men’s discus competition in the Summer
Olympics for 1908 to 1992.
extrapolation are less reliable than values
obtained by interpolation.
Discus Throw
4. A school principal wants to know if there is
y
a relationship between attendance and marks.
You have been hired to collect data and analyze
the results. You start by taking a sample
of 12 students.
64
56
48
40
0
1908
1912
1916
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
Distance (m)
72
x
Days
Absent
0
Average
(%)
93 79 81 87 87 75 77 90 77 72 61 80
Year
4
2
0
6
4
1
3
7
8
4
a) Create a scatter plot. Draw the line
of best fit.
b) What appears to be the average decrease
in marks for an absence of 1 day?
c) Predict the average of a student who
is absent for 6 days.
d) About how many days could a student
likely miss before getting an average
below 50%?
a) Estimate what the winning distance might
have been in 1940 and 1944.
b) Estimate the winning distance in 2020 and
2024.
2. As an object falls freely toward the ground, it
accelerates at a steady rate due to gravity. The
table shows the speed, or velocity, that an object
would reach at 1 s intervals during its fall.
3
5. A series of football punts is studied in an
experiment. The initial speed of the football
and the length of the punt are recorded.
Time from Start (s)
Velocity (m/s)
0
0
1
9.8
Speed (m/s)
10
17
18
21
25
2
19.6
Distance (m)
10
28
32
43
61
3
29.4
4
39.2
5
49.0
Use a curve of good fit to estimate the length of
a punt with an initial speed of 29 m/s, as well as
the initial speed of a punt with a length of 55 m.
a) Graph the data.
b) Determine the velocity of the object at 2.5 s,
3.5 s, and 4.75 s.
c) Estimate the velocity of the object obtained
at 6 s, 9 s, and 10 s.
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A–13 Transformations of Two-Dimensional Figures
Appendix A
A translation slides a figure up, down, right, left, or
diagonally along a straight line. For example, ^ABC has been
translated 1 unit right and 3 units down to create the image
triangle, ^A¿B¿C¿ .
B
A
B
C
A
C
y
8
A
A
6
E E
4
B
C
-6 -4
D
-2
Appendix B
A reflection flips a figure about a line of reflection to create
a mirror image. For example, pentagon ABCDE has been
reflected in the y-axis to create the image pentagon,
A¿B¿C¿D¿E¿. The y-axis is the line of reflection.
2
B
D
0
2
C
4
x
6
Glossary
EXAMPLE 1
Describe the transformations that have been performed on figures 1 and 2.
y
6
1
4
2
1
x
0
2
-2
6
Answers
-6
2
Solution
Figure 1 has been translated 9 units right and 2 units down.
Figure 2 has been reflected in the y-axis.
Index
EXAMPLE 2
^ ABC has vertices at A( - 5, 3), B( -2, 3), and C( -2, 1).
State the coordinates of ^ A¿B¿C¿ , if ^ ABC is
a) translated 8 units right and 3 units up
b) reflected in the x-axis
NEL
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Solution
a) When ^ ABC is translated 8 units right and
3 units up, the x-coordinate of each vertex
increases by 8 and the y-coordinate increases
by 3. The coordinates of the image are A¿(3, 6),
B¿ (6, 6), and C¿ (6, 4).
b) When ^ ABC is reflected in the x-axis, the
x-coordinates remain the same but the signs of
the y-coordinates change. The coordinates of the
image are A¿(-5, -3), B¿( -2, -3), and
C¿(- 2, - 1) .
y
8
A (3, 6)
6
B (6, 6)
A(–5, 3)
A(–5, 3) B(–2, 3)
C (6, 4)
x
C(–2, 1)
-8 -6 -4
-2
2
-2
4
6
B(–2, 3) 4
2
x
C(–2, 1)
C (–2, –1)
2
0
y
6
0
2
-2
A (–5, –3) B (–2, –3)
8
Practice
1. In what ways does the image differ from the
4. State the translation that was used
to create each image.
original when each transformation is performed?
a) translation
b) reflection
2. Identify the transformation.
a)
A
B
b) A
y
6
4
B
c)
2
x
D
C
CA
B
-4
C
-2
B
C
5. The coordinates of ^JKL are J(-4, 5),
K( -4, 2), and L(-1, 2). State the coordinates
of ^J¿K ¿L¿ , if ^JKL is
a) translated 2 units right, 5 units down
b) reflected in the x-axis
c) reflected in the y-axis
3. State the reflections that could be
performed on the flag to create each image.
a)
c)
b)
b)
a)
D
A
-2
0
d)
6. Describe how the figure was reflected to create
each image.
a)
y
b)
4
y
4
b)
x
-2 a)
-4
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0
2
-2
0
-2
x
2
-4
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A–14 Ratios, Rates, and Proportions
Appendix A
A ratio compares two or more quantities that are measured in the same
units. Because the units are the same, they are not included in the ratio.
3
A ratio can be written in three different forms: , 3 : 4, and 3 to 4.
4
A rate compares different quantities, so the units must be included.
Both rates and ratios can be simplified by dividing all the terms
by the greatest common factor.
Appendix B
When two ratios are equivalent, they can be used to write
a
c
a proportion: .
b
d
When one of the terms in a proportion is unknown, it can be found by
• using a multiplication or division relationship between the terms
in the numerators and denominators (inspection)
• using inverse operations to solve the equation (algebra)
EXAMPLE 2
In a class of 16 girls and 12 boys,
what is the ratio of boys to girls?
A bouquet of 25 flowers costs $20.
What is the cost per flower?
Solution
Solution
Boys to girls 12 : 16
12 16
:
4 4
3:4
Cost/flower = $20 , 25
$0.80/flower
Glossary
EXAMPLE 1
Answers
EXAMPLE 3
Determine x.
4
14
=
x
21
Solution
NEL
Solving Using Algebra
14
4
21
x
=
is the same as =
x
21
4
14
21
x
=
* 4
4 *
4
14
84
x =
14
x = 6
Appendix A: Review of Essential Skills and Knowledge
Index
Solving by Inspection
4
14
=
x
21
2
4
=
x
3
4 = 2 * 2, so
x = 2 * 3
x = 6
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Practice
1. Express each situation as a ratio.
6. Write the unknown term(s) in each proportion.
a) 2 : 3 = . : 6
b) 3 : 8 = . : 24
c) 8 : 5 = 16 : .
d) 1 : . : 8 = 3 : 12: e) 2 : 5 : . = 6 : : 9
a) 4 pucks to 5 sticks
b) 5 markers to 3 pens
c) 6 boys to 1 girl
d) 7 rabbits to 3 puppies
e) 4 elephants to 7 lions
2. Write each pair of quantities as a ratio, comparing
7. Solve each proportion. Evaluate your answers
to two decimal places, if necessary.
8
x
a)
=
14
40
36
10
b)
=
x
3
5
x
c)
=
7
95
55
13
d)
=
x
42
25
x
e)
=
12
65
18
x
f)
=
24
84
40
21
g)
=
x
35
152
x
h)
=
240
6
lesser to greater. Express the ratio in lowest terms.
a) 10 cm, 8 cm
b) 5 km, 3000 m
c) 30 s, 1.5 min
d) 400 g, 3 kg
e) 5 L, 2500 mL
f ) 40 weeks, 1 year
3. The scale for a scale diagram is 1 cm represents
2 m. Calculate the actual length for each length
on the scale diagram.
a) 2 cm
b) 9 cm
c) 12.25 cm
d) 24.2 cm
e) 13.5 cm
4. Express each situation as a rate.
a) Your heart beats 40 times in 30 s.
b) Your aunt drove 120 km in 1.5 h.
c) You bought 3.5 kg of ground beef
for $17.33.
d) You typed 360 words in 8 min.
5. Solve each proportion.
2
x
=
5
20
4
36
b)
=
x
7
9
24
c)
=
x
12
5
25
d)
=
x
2
a)
482
15
9
=
x
20
x
64
f)
=
15
24
16
20
g)
=
x
65
6
x
h)
=
7
21
8. The ratio of boys to girls at Highland Secondary
School is 3 to 4. If there are 435 boys at the
school, determine the total number of students
at the school.
9. A lawnmower engine requires oil and gas to be
e)
Appendix A: Review of Essential Skills and Knowledge
mixed in the ratio of 2 to 5. If John has 3.5 L
of gas in his can, how much oil should he add?
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A–15 Properties of Triangles and Angle Relationships
Complementary Angles
a + b = 90°
Appendix A
There are several special angle relationships.
Vertically Opposite Angles
a = b
c = d
Isosceles Triangle
∠ A = ∠C
AB = BC
B
a
Appendix B
c
d
b
b
a
A
Supplementary Angles
a + b = 180°
C
Sum of the Angles in a
Triangle
∠A + ∠B + ∠C = 180°
Exterior Angle of a Triangle
a + b = c
Glossary
A
b
a
b
a
B
c
C
Transversal and Parallel Lines
Alternate angles are equal.
c = f,d = g
Corresponding angles are equal.
b = f , a = e, d = h , c = g
NEL
f
c
Index
g
d
Co-interior angles are
supplementary.
d + f = 180°, c + e = 180°
b
a
c
Answers
The longest side is across from
the greatest angle. The shortest
side is across from the least
angle. In this triangle,
∠ A 7 ∠ B 7 ∠C , so
BC 7 AC 7 AB
d
f
e
g
f
d
h
c
e
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Practice
1. Determine the values of x, y, and z in each diagram.
a)
e)
y
y
80°
40°
z
x
z
x
60°
110°
f)
b)
50°
76°
50°
x
z
y
y
x
c)
z
g)
20°
x
x
y
z
y
120°
x
z
d)
50°
z
h)
y
y
110°
x
x
60°
z
60°
2. For each triangle, list the sides from longest to shortest, and the angles from greatest to least.
What do you notice?
a)
b) L
R
N
T
484
S
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M
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A–16 Congruent Figures
Appendix A
Two figures are congruent if they are the same size and shape. This
relationship is only true when all corresponding angles and all corresponding
sides are equal. The symbol for congruency is .
If ^ ABC ^ LMN , then the following is true.
A
Equal sides:
AB = LM
BC = MN
CA = NL
L
C M
Appendix B
B
Equal angles:
∠ABC = ∠ LMN
∠ BCA = ∠MNL
∠ CAB = ∠ NLM
N
EXAMPLE
Are quadrilaterals AMPJ and RWZE congruent? Give reasons for your answer.
61 cm
92°
70°
A
J
50 cm
48 cm
M
81°
46 cm
P
Glossary
117°
50 cm Z
81°
92°
46 cm
117°
61 cm
W
70°
48 cm
R
E
Solution
Answers
Use the diagram to determine whether the corresponding sides and angles
are equal.
Corresponding angles:
Corresponding sides:
∠ A = ∠ R, ∠J = ∠ E,
JA = ER, PM = WZ
∠ M = ∠ W, ∠ P = ∠ Z
AM = RW, JP = EZ
Quadrilateral AMPJ Quadrilateral RWZE
Practice
1. Draw two congruent rectangles on a coordinate
grid. Explain how you know that your rectangles
are congruent.
3. These two quadrilaterals are congruent.
Determine the values of x, y, and z.
R
2. Quadrilaterals JKLM and SPQR are congruent.
List all the pairs of equal sides and angles.
J
K
P
M
V
D
y 10
60°
K
R
L
M
7 cm
3x
S
Index
S
15 cm
B
45° W
12 z
Q
NEL
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Review of Technical Skills
Part 1 Using the TI–83/84 Graphing Calculator
B–1
Preparing the Calculator
487
B–2
Entering and Graphing a Relation
488
B–3
Evaluating a Relation
488
B–4
Changing the Window Settings
489
B–5
Using the Split Screen
489
B–6
Using the Table Feature
489
B–7
Making a Table of Differences
490
B–8
Determining the Zeros of a Relation
491
B–9
Determining the Maximum or Minimum Value of a Relation
491
B–10
Creating a Scatter Plot and Determining a Line or Curve
of Best Fit Using Regression
492
B–11
Determining the Points of Intersection of Two Relations
493
B–12
Evaluating Trigonometric Ratios and Determining Angles
494
B–13
Evaluating Powers and Roots
494
Part 2 Using The Geometer’s Sketchpad
B–14
Defining the Tool Buttons and Sketchpad Terminology
494
B–15
Selections in the Construct Menu
495
B–16
Graphing a Relation on a Cartesian Coordinate System
496
B–17
Creating and Animating a Parameter
497
B–18
Placing Points on a Cartesian Coordinate System Using
Plot Points
499
Placing Points on a Cartesian Coordinate System Using
the Point Tool
499
B–20
Determining the Coordinates of a Point
499
B–21
Constructing a Line, Segment, or Ray through a Given Point
500
B–22
Constructing and Labelling a Point on a Line, Segment, or Ray
500
B–23
Determining the Slope and Equation of a Line
501
B–24
Moving a Line
501
B–25
Constructing a Triangle and Labelling the Vertices
502
B–26
Measuring the Interior Angles of a Triangle
503
B–27
Constructing and Measuring an Exterior Angle of a Triangle
503
B–28
Determining the Sum of the Interior Angles of a Triangle
504
B–29
Measuring the Length of a Line Segment
505
B–30
Constructing the Midpoint of a Line Segment
506
B–31
Constructing the Perpendicular Bisector of a Line Segment
507
B–19
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B–32
Constructing the Bisector of an Angle
507
B–33
Measuring the Area of a Polygon
508
B–34
Constructing a Circle
509
Part 3 Using a Spreadsheet
B–35
510
Introduction to a Spreadsheet
Part 4 Using Fathom
B–36
Creating a Scatter Plot and Determining the Equation
of a Line or Curve of Good Fit
512
Appendix B
Part 5 Using the TI–nspire CAS and TI-nspire Handhelds
Beginning a New Document
513
B–38
Entering and Graphing a Relation
514
B–39
Evaluating a Relation
516
B–40
Changing the Window Settings
517
B–41
Using the Split Screen
518
B–42
Using the Function Table Feature
519
B–43
Making a Table of Differences
520
B–44
Determining the Zeros of a Relation
521
B–45
Determining the Maximum or Minimum Value of a Relation
522
B–46
Creating a Scatter Plot and Determining a Line or Curve
of Best Fit Using Regression
522
B–47
Determining the Points of Intersection of Two Relations
525
B–48
Evaluating Trigonometric Ratios and Determining Angles
526
B–49
Evaluating Powers and Roots
527
Glossary
B–37
Answers
PART 1 USING THE TI-83/84 GRAPHING
CALCULATOR
B–1
Preparing the Calculator
Before you graph a relation, be sure to clear any information left on the calculator
from the last time it was used. You should always do the following:
Index
1. Clear all data in the lists.
Press 2nd
4
ENTER
4
ENTER
1
.
2. Turn off all stat plots.
Press 2nd
Y=
.
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3. Clear all equations in the equation editor.
Y= , and then press
Press
CLEAR
for each equation.
4. Set the window so that the axes range from ⴚ10 to 10.
ZOOM
Press
B–2
. Press WINDOW to verify.
6
Entering and Graphing a Relation
1. Enter the equation of the relation in the equation editor.
To graph y ⫽ 2x ⫹ 8, press
GRAPH
Y=
2
X, T, U, n
1
8
. The graph will be displayed as shown.
2. Enter all linear equations in the form y ⫽ mx ⫹ b.
If m or b are fractions, enter them between brackets. For example, enter
2
7
2x ⫹ 3y ⫽ 7 in the form y = - x + , as shown. Make sure you use the
3
3
(2) key when entering negative numbers.
3. Press
GRAPH
to show the graph.
4. Press
TRACE
to determine the coordinates of any point on the graph.
Use
and
to cursor along the graph.
Press
ZOOM
ENTER
8
TRACE
to trace using integer
intervals. If you are working with several graphs at the same time,
use
B–3
and
(the up and down arrow keys) to scroll between graphs.
Evaluating a Relation
1. Enter the relation in the equation editor.
To enter y ⫽ 2x 2 ⫹ x ⫺ 3, press
X, T, U, n
2
3
Y=
2
X, T, U, n
x2
1
.
2. Use the value operation to evaluate the relation.
To determine the value of the relation at x ⫽ ⫺1, press 2nd
ENTER
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, enter (2)
1
at the cursor, and then press
TRACE
ENTER
.
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Changing the Window Settings
The window settings can be changed to show a graph on a different part
of the xy-axes.
1. Enter the relation in the equation editor.
For example, enter y ⫽ x 2 ⫺ 3x ⫹ 4 in the equation editor.
2. Use the Window function to set the boundaries for each axis.
To display the relation when - 2 … x … 5 and 0 … y … 14, press
1
B–5
ENTER
ENTER
, then
0
, then
1
ENTER
ENTER
3. Press
2
GRAPH
ENTER
ENTER
5
, then
, and finally
4
1
, then
, then
Appendix B
(2)
WINDOW
ENTER
1
.
to show the relation.
Using the Split Screen
Glossary
1. The split screen can be used to see a graph and the equation editor
at the same time.
Press
MODE
and cursor to Horiz. Press
then press 2nd
MODE
ENTER
to select this, and
to return to the home screen. Enter y ⫽ x 2 in
Y1 of the equation editor, and then press
GRAPH
.
2. The split screen can also be used to see a graph and a table at the same time.
MODE
ENTER
Answers
Press
, and move the cursor to G–T (Graph-Table). Press
to select this, and then press
GRAPH
. It is possible to view
the table with different increments. To find out how, see steps 2 and 3 in
Appendix B-6.
B–6
Using the Table Feature
A relation can be displayed in a table of values.
Index
1. Enter the relation in the equation editor.
To enter y ⫽ ⫺0.1x3 ⫹ 2x ⫹ 3, press
X, T, U, n
^
3
1
Y=
2
(2)
X, T, U, n
1
.
1
3
.
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2. Set the start point and step size for the table.
Press 2nd
WINDOW . The cursor is beside TblStart⫽. To start at
x ⫽ ⫺5, press (2)
ENTER
5
. The cursor is now beside
⌬Tblⴝ (⌬, the Greek capital letter delta, stands for change in). To increase
ENTER
1
the x-value by 1s, press
GRAPH
3. To view the table, press 2nd
Use
and
.
.
to move up and down the table. Notice that you can look
at greater or lesser x-values than those in the original range.
B–7
Making a Table of Differences
To make a table with the first and second differences for a relation, use the STAT lists.
1. Press STAT
1
, and enter the x-values into L1.
For y ⫽ 3x 2 ⫺ 4x ⫹ 1, use x-values from ⫺2 to 4. Input each number
ENTER
followed by
.
2. Enter the relation.
Scroll right and up to select L2. Enter the relation y ⫽ 3x 2 ⫺ 4x ⫹ 1, using
L1 as the variable x. Press
4
2
3. Press
2nd
ENTER
1
ALPHA
1
3
1
1
ALPHA
2nd
x2
1
1
.
to display the values of the relation in L2.
4. Determine the first differences.
Scroll right and up to select L3. Then press 2nd
STAT . Scroll right to
OPS and press
7
to choose ¢ List(. Enter L2 by pressing
2nd
)
. Press
2
ENTER
to see the first differences
displayed in L3.
5. Determine the second differences.
Scroll right and up to select L4. Repeat step 4, using L3 instead of L2. Press
ENTER
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Appendix B: Review of Technical Skills
to see the second differences displayed in L4.
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Determining the Zeros of a Relation
To determine the zeros of a relation, use the zero operation.
1. Start by entering the relation in the equation editor.
For example, enter y ⫽ ⫺(x ⫹ 3)(x ⫺ 5) in the equation editor. Then press
GRAPH
ZOOM
6
.
2
.
2. Access the zero operation.
TRACE
3. Use
and
of the right zero.
ENTER
Press
Appendix B
Press 2nd
to cursor along the curve to any point that is left
to set the left bound.
4. Cursor along the curve to any point that is right of the right zero.
ENTER
Press
to set the right bound.
Glossary
ENTER again to display the coordinates of the zero
(the x-intercept).
5. Press
6. Repeat to determine the coordinates of the left zero.
B–9
Answers
Determining the Maximum or Minimum
Value of a Relation
The least or greatest value can be found using the Minimum operation or
the Maximum operation.
1. Enter and graph the relation.
For example, enter y ⫽ ⫺2x 2 ⫺ 12x ⫹ 30. Graph the relation, and adjust
the window to get a graph like the one shown. This graph opens downward,
so it has a maximum.
2. Use the maximum operation.
2nd
4
TRACE
TRACE
3
Index
Press 2nd
. For parabolas that open upward, press
to use the Minimum operation.
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3. Use
and
to cursor along the curve to any point that is left
of the maximum value.
Press
ENTER
to set the left bound.
4. Cursor along the curve to any point that is right of the maximum value.
Press
ENTER
to set the right bound.
5. Press
ENTER
again to display the coordinates of the maximum value.
B–10 Creating a Scatter Plot and Determining
a Line or Curve of Best Fit Using Regression
This table gives the height of a baseball above ground, from the time it was hit
to the time it touched the ground.
Time (s)
0
1
2
3
4
5
6
Height (m)
2
27
42
48
43
29
5
You can use a graphing calculator to create a scatter plot of the data.
1. Start by entering the data into lists.
ENTER
Press STAT
. Move the cursor over to the first position in L1,
and enter the values for time. Press
ENTER
after each value. Repeat this
for height in L2.
2. Create a scatter plot.
Press 2nd
Y=
and
1
ENTER
. Turn on Plot 1 by making sure
that the cursor is over On, the Type is set to the graph type you prefer, and L1
and L2 appear after Xlist and Ylist.
3. Display the graph.
Press
492
Appendix B: Review of Technical Skills
ZOOM
9
to activate ZoomStat.
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4. Apply the appropriate regression analysis.
To determine the equation of the line or curve of best fit, press STAT and
5
scroll over to CALC. In this case, press
4
you need a line of best fit, press
Then press 2nd
,
1
1
to Y-VARS. Press
to enable QuadReg. (When
to enable LinReg(ax⫹b).
2nd
2
,
VARS . Scroll over
twice. This action stores the equation of the line or
Appendix B
curve of best fit into Y1 of the equation editor.
5. Display and analyze the results.
Press
ENTER
. In this example, the letters a, b, and c are the coefficients
of the general quadratic equation y ⫽ ax2 ⫹ bx ⫹ c for the curve of best fit.
R 2 is the percent of data variation represented by the model. The equation
is about y ⫽ ⫺4.90x2 ⫹ 29.93x ⫹ 1.98.
Note: For linear regression, if r is not displayed, turn on the diagnostics
ENTER
0
Glossary
function. Press 2nd
, and scroll down to DiagnosticOn. Press
twice. Repeat steps 4 and 5.
6. Plot the curve.
Press
GRAPH
.
Answers
B–11 Determining the Points of Intersection
of Two Relations
1. Enter both relations in the equation editor.
For example, enter y ⫽ 5x ⫹ 4 into Y1 and y ⫽ ⫺2x ⫹ 18 into Y2.
2. Graph both relations.
Press GRAPH
is displayed.
. Adjust the window settings until the point of intersection
3. Use the intersect operation.
TRACE
5
Index
Press 2nd
.
4. Determine a point of intersection.
You will be asked to verify the two curves and enter a guess (optional) for the
point of intersection. Press
ENTER
after each screen appears. The point
of intersection is exactly (2, 14).
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B–12 Evaluating Trigonometric Ratios
and Determining Angles
1. Put the calculator in degree mode.
Press
MODE
ENTER
. Scroll down and across to Degree. Press
.
2. Use the SIN , COS , or TAN key to calculate a trigonometric ratio.
To determine the value of sin 54°, press SIN
ENTER
5
)
4
.
3. Use SINⴚ1, COSⴚ1, or TANⴚ1 to calculate an angle.
To determine the angle whose cosine is 0.6, press 2nd
)
ENTER .
6
COS
.
B–13 Evaluating Powers and Roots
1. Evaluate the power 5.32.
3
x2
ENTER
5
Press 7
.
-2
3. Evaluate the power 8 .
^
5
ENTER
(2)
2
ENTER
.
Press
5
.
.
2. Evaluate the power 7.55.
Press
8
^
.
4. Evaluate the square root of 46.1
Press 2nd
x2
4
6
.
1
)
ENTER
.
PART 2 USING THE GEOMETER’S SKETCHPAD
B–14 Defining the Tool Buttons and Sketchpad
Terminology
Sketches and Dynamic Geometry
The ability to change objects dynamically is the most important feature of The
Geometer’s Sketchpad. Once you have created an object, you can move it, rotate it,
dilate it, reflect it, or hide it. You can also change its label, colour, shade, or line
thickness. No matter what changes you make, The Geometer’s Sketchpad maintains
the mathematical relationships between your object and other objects it is related
to. This is the principle of dynamic geometry and the basis of the power and
usefulness of The Geometer’s Sketchpad.
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Sketchpad Terminology
menu bar
Select.
Construct a point.
Construct a circle.
Construct a line.
Label.
Custom Tool.
Appendix B
toolbar
display area
Select means move the mouse pointer to the desired location and click the mouse
button (left-click for Windows users).
Deselect means select the selection tool and click anywhere in the display area,
away from any figures you have drawn.
Drag means move the mouse pointer to the point or figure you would like to
move. Click on the point or figure and, while holding down the mouse button,
move the point or figure to a new location. Release the mouse button when the
point or figure is in the desired position. This can also be done to text and labels.
B–15 Selections in the Construct Menu
Command
What It Constructs
What You Must Select
Point On Object
a point on the selected object(s)
one or more segments, rays, lines, or circles
Intersection
a point where two objects intersect
two straight objects, two circles, or a
straight object and a circle
Midpoint
the midpoint of the segment(s)
one or more segments
Segment/Ray/Line
the segment(s), ray(s), or line(s)
defined by the points
two or more points
Parallel Line
the line(s) through the selected
point(s), parallel to the selected
straight object(s)
one point and one or more straight
objects, or one straight object and one or
more points
Perpendicular Line
the line(s) through the selected
point(s), perpendicular to the selected
straight object(s)
one point and one or more straight
objects, or one straight object and one or
more points
Angle Bisector
the ray that bisects the angle defined
by three points
three points (select the vertex second)
Circle By Centre
ⴙ Point
the circle with the given centre and
passing through the given point
two points (select the centre first)
(continued)
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Command
What It Constructs
What You Must Select
Circle By Centre
ⴙ Radius
the circle with the given centre and
with a radius equal to the length of
the given segment
a point and a segment
Arc On Circle
the arc that extends counterclockwise
from the first point on a circle to the
second point
a circle and two points on the circumference
of the circle
Arc Through
3 Points
the arc that passes through the three
given points
three points
Polygon Interior
the interior of a polygon defined by
using the given points as its vertices
three or more points
Circle Interior
the interior of a circle
one or more circles
Sector Interior
the interior of an arc sector
one or more arcs
Arc Segment
Interior
the interior of an arc segment
one or more arcs
Locus
the locus of an object
one geometric object and one point
constructed to lie on a path
Note: At any point in time, your selections determine which menu commands are
available at that point. When a command is not available, it is greyed out in its
menu. Often this means your current selection is not appropriate for that command.
B–16 Graphing a Relation on a Cartesian
Coordinate System
You can graph relations on a Cartesian coordinate
system in The Geometer’s Sketchpad. For example, use
the following steps to graph the relation y = 2x + 8.
1. Turn on the grid.
From the Graph menu, choose Define
Coordinate System.
2. Enter the relation.
From the Graph menu, choose Plot New
Function. The New Function calculator should
appear. Use either the calculator keypad or the
keyboard to enter 2 * x ⫹ 8.
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3. Graph the relation y = 2x + 8.
Press OK on the calculator keypad. The graph
of y = 2x + 8 should appear on the grid.
Appendix B
4. Adjust the origin and/or the scale.
To adjust the origin, click on the point at the
origin to select it. Then click and drag the origin
as desired.
To adjust the scale, click in blank space to deselect.
Then click on the point at (1, 0) to select it.
Click and drag this point to change the scale.
B–17 Creating and Animating a Parameter
You can control relations in The Geometer’s Sketchpad
using parameters. For example, follow these steps to
graph the relation y = ax 2 using a as the parameter.
1. Turn on the grid.
2. Create the parameter.
From the Graph menu, choose New
Parameter.… Enter the Name of the parameter:
a. Change the Value to 3.0, and leave none as
the Units selected.
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3. Click OK.
The parameter will appear on the screen, in the same
format as a label.
4. Enter the relation.
From the Graph menu, choose Plot New Function.
The New Function calculator will appear. Click on
the parameter. The letter a will appear on the
calculator screen. Then use either the calculator
keypad or the keyboard to enter *x^2. Since the value
of the parameter is 3, the relation plotted is y = 3x 2.
Click OK.
5. Change the parameter.
Try a = 2. Use the Selection Tool to double-click on
the parameter, and enter 2. Try other values
for a.
6. Animate the parameter.
Use the Selection Tool to select only parameter a.
From the Display menu, choose Animate Parameter.
The Motion Controller will appear. Use the
,
,
, and
controls to start, stop,
reverse, and pause. Use the Speed box to change
the speed.
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B–18 Placing Points on a Cartesian Coordinate
System Using Plot Points
Sometimes, you may want to plot points without
graphing a relation. For example, suppose that you
want to plot (2, 1), (3, 5), and (–2, 0).
1. Turn on the grid.
2. Enter the coordinates of the points.
Appendix B
From the Graph menu, select Plot Points ….
For each point you want to plot, enter the
x-coordinate followed by the y-coordinate. Use
the Tab key on your keyboard to move from one
coordinate entry space to the next. Select Plot
when you have entered both coordinates of
a point. You can continue entering coordinates
until you click on Done.
B–19 Placing Points on a Cartesian Coordinate
System Using the Point Tool
You can also use the Point Tool to plot points without
graphing a relation. For example, suppose that you
want to plot (4, 2).
1. Turn on the grid.
2. Select the Point Tool.
The selection arrow will now look like a dot. This
indicates that when you click on the grid, a point
will be placed at the location you clicked.
B–20 Determining the Coordinates of a Point
1. Turn on the grid.
2. Plot some points on the grid.
3. Use the Selection Tool to select a point.
4. From the Measure menu, select Coordinates.
The coordinates of the selected point(s) will be
displayed.
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B–21 Constructing a Line, Segment, or Ray
through a Given Point
1. Turn on the grid.
2. Plot the point you want the line (or segment or
ray) to pass through.
3. Plot a second point anywhere on the grid.
4. Shift-click to make sure that both points
are selected.
If you are constructing a ray, make sure that
the point where the ray begins is selected first.
5. From the Construct menu, select Line
(or Segment or Ray).
B–22 Constructing and Labelling a Point on
a Line, Segment, or Ray
1. Turn on the grid.
2. Draw a line (or segment or ray).
3. Select the line by clicking on it.
4. From the Construct menu, select
Point On Line.
5. Select the Label Tool. Use it to double-click on
the point you constructed.
A label will appear beside the point, as well as a
Properties box for the point. You can change the
label of the point by changing the contents of
the label entry in the Properties box.
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B–23 Determining the Slope and Equation of a Line
1. Turn on the grid.
2. Draw a line.
3. Use the Selection Tool to select the line.
4. From the Measure menu, select Slope or
Equation.
Appendix B
B–24 Moving a Line
1. Turn on the grid.
2. Draw a line.
3. Copy the line by clicking Copy and then
Paste from the Edit menu.
4. To keep the new line parallel to the
original line, follow the steps below:
Use the Selection Tool to click and hold only
the line. Hold the mouse button down while
you move the mouse. The new line will move
parallel to the original line.
5. To move the new line so that one point stays
in the same position, follow these steps:
Use the Selection Tool to select a point on the
line, other than the point that is staying in the
same position. Hold the mouse button down
while you move the mouse. The line will move
as the mouse moves, but the original point will
stay fixed.
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B–25 Constructing a Triangle and Labelling
the Vertices
1. Open a new sketch.
2. Use the Point Tool to place three points.
If you hold down the Shift key on your keyboard
while you place the points, all the points will
remain selected as you place them.
3. From the Display menu, select Show Labels.
The order in which you select the points
determines the alphabetical order of the labels.
4. From the Construct menu, select Segments.
The three sides of the triangle will be displayed.
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B–26 Measuring the Interior Angles of a Triangle
1. Open a new sketch, and draw a triangle with
the vertex labels displayed.
2. Shift-click to select the vertices that form
an angle.
For example, to measure ∠ ABC, select
vertex A, then vertex B, and then vertex C.
3. From the Measure menu, select Angle.
Appendix B
4. Repeat steps 2 and 3 for each angle.
B–27 Constructing and Measuring an Exterior
Angle of a Triangle
1. Open a new sketch, and draw a triangle
with the vertex labels displayed.
2. Select two vertices. From the Construct
menu, select Ray.
This will extend one side of the triangle to
form an exterior angle.
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3. While the ray is selected, choose Point On Ray
from the Construct menu.
4. Drag the point so that it is outside the triangle.
Display the label for the point.
5. Select the exterior angle.
Select the point, then the vertex for the angle,
and then the final vertex. From the Measure
menu, select Angle.
B–28 Determining the Sum of the Interior Angles
of a Triangle
1. Open a new sketch, and draw a labelled triangle.
Measure all three interior angles.
Shift-click to select all three angle measures.
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2. From the Measure menu, select Calculate.
A New Calculation window will appear.
Appendix B
3. Use the Values pop-up menu to create the
formula for the sum of the selected angles.
Choose an angle from the Values pop-up.
Then enter ⫹ and continue entering addends
until the formula is complete.
Click OK when you are finished.
B–29 Measuring the Length of a Line Segment
1. Open a new sketch, and draw a line
segment.
2. While the line segment is selected, choose
Length from the Measure menu.
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The length will be displayed.
B–30 Constructing the Midpoint of a Line Segment
1. Open a new sketch, and draw a line segment.
2. While the line segment is selected, choose
Midpoint from the Construct menu.
The midpoint will be displayed.
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B–31 Constructing the Perpendicular Bisector
of a Line Segment
1. Open a new sketch, and draw a line
segment.
2. While the line segment is selected, choose
Midpoint from the Construct menu.
3. Select the segment and the midpoint.
4. From the Construct menu, choose
Perpendicular Line.
Appendix B
The perpendicular bisector will be displayed
as a line through the selected point.
B–32 Constructing the Bisector of an Angle
1. Open a new sketch, and place three points
to form an angle.
2. Use the Ray Tool or the Segment Tool to
draw the angle.
3. Select the vertices that form the angle.
4. From the Construct menu, choose
Angle Bisector.
The angle bisector will be displayed as a ray
going out from the angle.
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B–33 Measuring the Area of a Polygon
1. Open a new sketch. Place points for the
vertices of a polygon and their labels.
Make sure that the vertices are placed and selected
in order in either a clockwise or counterclockwise
direction.
2. While the points are selected, use the Construct
Polygon Interior operation in the Construct
menu to form a polygon.
The Geometer’s Sketchpad will name the polygon
depending on the number of vertices you have
selected.
3. While the interior is highlighted, select Area
from the Measure menu.
The area will be displayed.
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B–34 Constructing a Circle
1. Open a new sketch.
2. Use the Circle Tool to select a point
for the centre of the circle.
To change the size of the circle, hold the mouse
button down and drag the point on the
circumference.
Appendix B
3. Turn on the grid.
To construct a circle with a given centre, passing
through a given point, follow either step 4 or
step 5 below.
4. Using the Circle Tool, click on the centre to
select it.
Place the cursor at the origin. Drag the mouse
to the given point on the circle, and click again.
If your point has integer coordinates, select
Snap Points from the Graph menu before you
begin to drag.
5. Using the Point Tool, draw a point for the
centre and another point to be on the
circumference.
Select the centre and then the point. From the
Construct menu, choose Circle By
CenterⴙPoint.
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PART 3 USING A SPREADSHEET
B–35 Introduction to a Spreadsheet
A spreadsheet is a computer program that can be used to create a table of values
(from data or an equation) and then graph the values as a scatter plot. A spreadsheet
can also be used, when necessary, to determine the equation of the line or curve of
best fit using regression.
A spreadsheet consists of cells that are identified by column letter and row number,
such as A2 or B5. A cell can hold a label, a number, or a formula.
The following table of values shows the time and height of a ball that was thrown
into the air.
Time (s)
0
1
2
3
4
5
Height (m)
2.0
22.1
32.4
32.9
23.6
4.5
Note: Different spreadsheets have different commands. Check the instructions for
your spreadsheet to determine the proper commands to use. The instructions below
are for some versions of Microsoft Excel.
Entering Data to Create a Table
To create a spreadsheet for the data above, open the program to create a new
worksheet. Label cell A1 “time (s)” and cell B1 “height (m).”
Enter the initial values. Enter 0 in A2 and 2.0 in B2. Repeat this process to enter
the rest of the time data in column A and the corresponding heights in column B,
in rows 3 to 7.
Creating a Scatter Plot
Use the cursor to highlight the part of the table that you want to graph.
For this example, select columns A and B in rows 1 to 7. Use the
spreadsheet’s graphing command (
for this program)
to graph the data. A scatter plot like this will appear. Notice that axes labels
are included.
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Performing a Regression
Select any point on the scatter plot, and right click. Select Add
Trendline… in the menu that pops up (or go to Chart in the Menu Bar
and select Add Trendline…). The window at the right will appear.
Appendix B
There are several different types of regressions that are available. (Those
that are grey are not applicable to the current data.) Linear is the default.
In this example, Polynomial of Order 2 (quadratic regression) is used.
Select the Options tab, and you will see several properties that might be
useful. The Forecast option allows you to graph the trendline beyond the
data. (This is useful when extrapolating values.) The y-intercept can be
forced to be a particular value if the value is known. To see the model
equation for your regression analysis, select Display equation on chart.
Click OK. A parabola, graphed with the scatter plot, and the equation of
the parabola will appear.
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PART 4 USING FATHOM
B–36 Creating a Scatter Plot and Determining the
Equation of a Line or Curve of Good Fit
1. Create a case table.
Drag a case table from the object shelf, and drop it in the document.
2. Enter the variables and data.
Click on <new>, type a name for the new variable or attribute, and press
Enter on your keyboard. (If necessary, repeat this step to add more attributes.
Pressing Tab instead of Enter moves you to the next column.) When you
name your first attribute, Fathom creates an empty collection to hold your
data (a little empty box). This is where your data are actually stored. Deleting
the collection deletes your data. When you add cases by typing values, the
collection icon fills with gold balls. To enter the data, click in the blank cell
under the attribute name and begin typing values. (Press Tab to move from
cell to cell.)
3. Graph the data.
Drag a new graph from the object shelf at the top of the
Fathom window, and drop it in a blank space in your
document. Drag an attribute from the case table, and drop it
on the prompt below and/or to the left of the appropriate axis
in the graph.
4. Create a relation.
Right-click on the graph, and select Plot Function. Enter the right
side of your relation using a parameter that can be adjusted to fit the
curve to the scatter plot (a was used below). In this case in the
equation, height was used for the dependent variable and x was used
for the independent variable. Time could also have been used as the
independent variable.
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5. Create a slider for the parameter(s) in your equation.
Drag a new slider from the object shelf at the top of the Fathom
window, and drop it in a blank space below your graph. Over V1,
type the letter of the parameter used in your relation in step 4.
Click on the number, and then adjust the value of the slider until
you are satisfied with the fit. This can be done by moving the
cursor over the slider’s number line and clicking and dragging to
adjust, and dragging the slider as needed.
The equation of a curve of good fit is y = - 4.8(x + 0.2)(x - 6.2).
Appendix B
PART 5 USING THE TI-nspire CAS AND
TI-nspire HANDHELDS
B–37 Beginning a New Document
To begin a new document, you should always do the following:
1. Select a New Document.
and scroll to 6: New Document. Press
.
Glossary
Press
Answers
2. Save the previous document.
If you want to save the previous document, select Yes and then
press
. If you do not want to save the previous document,
select No and then press
using either the
. (You can move the cursor by
key or the and keys.) Selecting No
Index
closes all applications that are open in the document.
Appendix B: Review of Technical Skills
513
3. Select the application.
Select the application you want to use, and then press
.
(You can move the cursor by using the and keys.) Every time
you add a new application, it appears as a numbered page in your
document. To move between these applications, press
and use
either the or key to scroll to the desired application.
B–38 Entering and Graphing a Relation
Enter the equation of the relation into the Graphs & Geometry data entry line.
The handheld will display the graph.
1. Select 2: Graphs & Geometry from the application menu.
2. Graph.
To show the scale on the graph, press
and scroll to
2: View 8: Show Axes End Values, and press
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3. Enter all equations in the form y = mx + b.
7
2
For example, for 2x + 3y = 7, enter - x + in the data
3
3
entry line. Do so by pressing
.
Appendix B
4. Press
to view the graph.
To move the cursor from the data entry line to the graph,
press
twice.
Glossary
5. To determine the coordinates of any point on the graph,
press
.
Answers
Scroll to 5: Trace 1: Graph Trace, and press
. A point will
appear on the graph. Use the and keys to move the cursor along
the graph. Press
to stop the Trace feature.
Index
If you are working with several graphs at the same time, a trace point
will appear on all relations that have a point in the window shown.
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515
B–39 Evaluating a Relation
1. Enter the relation into the data entry line of the Graphs & Geometry
application.
To enter y = 2x 2 + x - 3, press
Press
.
to display the graph.
2. Use the Points & Lines menu to evaluate the relation.
To determine the value of the relation at x = - 1, press
.
Scroll to 6: Points & Lines 2: Point On and press
to place
a point on the relation.
Move the cursor/pencil to any location on the graph. Press
to
plot the point at this location. The point does not have to be located
at the value you really want. Press
of the point until it is flashing. Press
Press
, then hover over the x-value
to select the x-value.
as many times as necessary to delete all the digits
of the x-value, then enter ⫺1.
Press
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to see the value of the relation at x = - 1.
3. Determine when a relation has a specific y-value.
Hover over the value of the y-coordinate of any point to change it to the required
y-value. For example, to determine when the relation is equal to 3, enter 3 for the
y-value.
Appendix B
The relation has a y-value of 3 when x = - 2.
To determine the other x-value when the y-value is 3, move the point (or plot a new
point) closer to this location, and repeat the previous procedure.
Glossary
B–40 Changing the Window Settings
The window settings can be changed to show a graph on a different part
of the x y-axes.
1. Enter the relation y = x 2 - 3x + 4 into the data entry line of the
Graphs & Geometry application.
2. Select the Window feature to set the boundaries of the graph.
Answers
To display the relation when - 2 … x … 5 and 0 … y … 14,
press
, scroll to 4: Window 1: Window Settings, and
press
. Press
, then
,
then
, then
, then
then
, and finally
. The graph will appear as
,
Index
shown on the next page.
Appendix B: Review of Technical Skills
517
B–41 Using the Split Screen
To see a graph and a table at the same time: press
, then scroll to
5: Page Layout 2: Select Layout 2: Layout 2 (your choice) and press
518
.
Press
followed by
Press
to select the Lists & Spreadsheet application, and press
Press
and scroll to 5: Function Table 1: Switch to Function Table.
Press
to select the choice, then press
Appendix B: Review of Technical Skills
to move the cursor to the empty screen.
.
again to carry out the request.
B–42 Using the Function Table Feature
A relation can be displayed in a table of values.
1. Enter the relation into the data entry line of the Graphs & Geometry
application.
For example, to enter the relation y = - 0.1x 2 + 2x + 3, press
.
2. Add the Lists & Spreadsheet application.
Press
, and scroll to 3: Lists & Spreadsheet application. Press
.
Appendix B
3. View the function table.
Press
, and scroll to 5: Function Table 1: Switch to Function Table.
Press
to select the choice, and then press
again to carry out the request.
Glossary
4. Set the start point and step size for the table.
the table start at x = - 3 and increase in increments of 0.5, press
,
scroll to 5: Function Table 3: Edit Function Table Settings and press
and then adjust the settings as shown. Then
Answers
It is possible to view the table with different increments. For example, to see
to OK and press
,
.
Index
Appendix B: Review of Technical Skills
519
B–43 Making a Table of Differences
To create a table with the first and second differences for a relation,
add a List & Spreadsheet application.
1. Add the Lists & Spreadsheet application.
Press
, scroll to 3: Lists & Spreadsheet application, then press
.
For the relation y = 3x 2 - 4x + 1, enter x-values from ⫺2 to 4 into
column A.
2. Enter the relation.
Scroll right and up to select the shaded formula cell for column B.
Enter the relation, using
as the variable x. Press
.
3. Press
to display the values of the relation in column B.
4. Determine the first differences.
Scroll right and up to the shaded formula cell for column C. Then press
. Scroll down until you can choose ¢ List(.
Press
, then
Press
to see the
first differences displayed
in column C.
520
Appendix B: Review of Technical Skills
.
5. Determine the second differences.
Scroll right and up to select the shaded formula cell for column D.
Repeat step 4, using
instead of
. Press
to show
the second differences displayed in column D.
Appendix B
B–44 Determining the Zeros of a Relation
To determine the zeros of a relation, use the Trace feature in the
Graphs & Geometry application.
1. Enter the relation.
Enter y = - (x + 3)(x - 5) into the data entry line
Glossary
of the Graphs & Geometry application. Press
.
2. Access the Trace feature.
Press
press
and scroll to 5: Trace 1: Graph Trace, then
.
Answers
3. Use the and keys to move the cursor along the curve.
A point will appear on the graph. Using the cursor, move the point until
the word zero is displayed. Repeat to determine the second zero.
Index
Appendix B: Review of Technical Skills
521
B–45 Determining the Maximum or Minimum
Value of a Relation
The least or greatest value can be found using the Trace feature in the
Graphs & Geometry application.
1. Enter y = - 2x 2 - 12x + 30.
Graph the relation and adjust the window as shown. The graph opens
downward, so it has a maximum.
2. Access the Trace feature.
Press
and scroll to 5: Trace 1: Graph Trace, then press
.
3. Use the and keys to move the cursor along the curve.
A point will appear on the graph. Move the point until the word
maximum is displayed. If a graph has a minimum, the point will be
displayed with the word minimum.
B–46 Creating a Scatter Plot and Determining a
Line or Curve of Best Fit Using Regression
This table gives the height of a baseball above the ground, from the time it was hit
to the time it touched the ground.
Time (s)
0
1
2
3
4
5
6
Height (m)
2
27
42
48
43
29
5
Create a scatter plot of the data:
1. Enter the data into the Lists & Spreadsheet application.
Move the cursor to the top cell of column A, and enter the word time. Enter
the values for time starting in row 1. Press
for height in column B.
522
Appendix B: Review of Technical Skills
after each value. Repeat this
If you need to resize the width of the column, press
and scroll to
1: Actions 2: Resize 1: Resize Column Width. Use the key to make the
column wider. Press
, then
when the column is wide enough.
Appendix B
2. Create a scatter plot.
to add a Graphs & Geometry application.
Press
and scroll to 3: Graph Type 4: Scatter Plot.
Press
Glossary
Press
.
to select time. Press
again. Press
, then use the arrow keys
Answers
To select time for the x-value, press
to move to y.
Repeat this process to select height.
Index
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3. Display the graph.
Press
and scroll to 4: Window 9: Zoom – Data. Press
.
4. Apply the appropriate regression analysis.
To determine the equation of the line or curve of best fit, return to the
List & Spreadsheet application.
Press
. Move the cursor to the first cell in an empty column. Press
and scroll to 4: Statistics 1: Stat Calculations . Select the appropriate
regression. In this case, select 6: Quadratic Regression and press
Use the
key to select time for the X List, then press
to move to the Y List, use the
524
. Press
key to select height for the
Y List, then press
. Continue to press
highlighted. Press
.
Appendix B: Review of Technical Skills
.
until OK is
5. Display and analyze the results.
In this example, the letters a, b, and c are the coefficients of the
general quadratic equation y = ax 2 + bx + c for the curve of
best fit. R 2 is the percent of data variation represented by the
model. The equation is about y = - 4.90x 2 + 29.93x + 1.98.
6. Plot the curve.
Return to the Graphs & Geometry application. Press
.
The graph type is currently set for a scatter plot. The regression
and scroll to 3: Graph Type
1: Function, and then press
. The regression equation is stored in f1(x).
Press
to see what is written in f1(x). Press
Appendix B
equation is a relation. Press
to show the graph.
Glossary
Answers
B–47 Determining the Points of Intersection of
Two Relations
1. Enter both relations into the data entry line of the Graphs & Geometry
application.
For example, enter y = 5x + 4 in f1(x), and then press
Enter y = - 2x + 18 in f2(x), and then press
.
.
2. Graph both relations.
empty area in the window. Hold the click
Index
Adjust the window settings until the point(s) of intersection is
(are) displayed. You can do this by changing the window settings
or by picking up the graph and moving it. Move the cursor to an
key down until a
closed fist appears. Then use the arrow keys to move the graph
until you see the intersection point(s).
Appendix B: Review of Technical Skills
525
3. Use the intersection feature.
Press
and scroll to 6: Points & Lines 3: Intersection
Point(s), and then press
.
4. Determine a point of intersection
You will need to select the two lines. Move the cursor to one of
the lines. When the line blinks, press
to select the line.
Repeat this process to select the second line.
As you move the cursor near the second line, a point will appear at the
intersection. To make the point permanent and to know its coordinates,
press
. The point of intersection is (2, 14).
Note: If there is more than one point of intersection, all the points of
intersection that are visible in the window will appear.
B–48 Evaluating Trigonometric Ratios and
Determining Angles
1. Put the handheld in degree mode.
526
Press
. Select 8: System Info 1: Document Settings.
Use
to move through the selections. At Angle select Degree
using the
key.
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to select Degree. Continue to press
Then press
2. Use the
3:42 PM
. Press
,
until OK is selected.
and select the calculator application.
, or
key to calculate trigonometric
ratios.
To determine the value of sin 45°, press
. The answer will be exact. To determine the decimal
, then
.
Appendix B
approximation, press
3. Use SINⴚ1, COSⴚ1, or TANⴚ1 to calculate angles.
To determine the angle whose cosine is 0.6, press
.
Glossary
B–49 Evaluating Powers and Roots
Answers
Use the calculator application.
1. Evaluate the power 5.32.
Press
.
2. Evaluate the power 7.55.
Press
.
Press
Index
3. Evaluate the power 8ⴚ2.
.
4. Evaluate the square root of 46.1.
Press
.
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Glossary
Instructional Words
C
calculate: Figure out the number that answers a question.
Compute.
clarify: Make a statement easier to understand. Provide
an example.
classify: Put things into groups according to a rule and
label the groups. Organize into categories.
compare: Look at two or more objects or numbers and
identify how they are the same and how they are different
(e.g., compare the numbers 6.5 and 5.6; compare the size
of the students’ feet; compare two figures).
conclude: Judge or decide after reflection or after
considering data.
construct: Make or build a model. Draw an accurate
geometric figure (e.g., use a ruler and a protractor to
construct an angle).
create: Make your own example.
D
describe: Tell, draw, or write about what something is
or what something looks like. Tell about a process in a
step-by-step way.
determine: Decide with certainty as a result of
calculation, experiment, or exploration.
extend: 1. In patterning, continue the pattern.
2. In problem solving, create a new problem that takes the
idea of the original problem further.
J
justify: Give convincing reasons for a prediction, an
estimate, or a solution. Tell why you think your answer
is correct.
M
measure: Use a tool to describe an object or determine an
amount (e.g., use a ruler to measure the height or distance
around something; use a protractor to measure an angle;
use balance scales to measure mass; use a measuring cup to
measure capacity; use a stopwatch to measure the time in
seconds or minutes).
model: Show or demonstrate an idea using objects
and/or pictures (e.g., model addition of integers using red
and blue counters).
P
predict: Use what you know to work out what is going to
happen (e.g., predict the next number in the pattern 1, 2,
4, 7, … ).
draw: 1. Show something in picture form (e.g., draw a
diagram).
2. Pull or select an object (e.g., draw a card from the deck;
draw a tile from the bag).
R
E
relate: Describe how two or more objects, drawings,
ideas, or numbers are similar.
estimate: Use your knowledge to make a sensible
decision about an amount. Make a reasonable guess
(e.g., estimate how long it takes to cycle from your home
to school; estimate how many leaves are on a tree; what is
your estimate of 3210 ⫹ 789?).
evaluate: 1. Determine if something makes sense. Judge.
2. Calculate the value as a number.
explain: Tell what you did. Show your mathematical
thinking at every stage. Show how you know.
explore: Investigate a problem by questioning,
brainstorming, and trying new ideas.
528
Glossary
reason: Develop ideas and relate them to the purpose of
the task and to each other. Analyze relevant information
to show understanding.
represent: Show information or an idea in a different way
that makes it easier to understand (e.g., draw a graph;
make a model).
S
show (your work): Record all calculations, drawings,
numbers, words, or symbols that make up the solution.
sketch: Make a rough drawing (e.g., sketch a picture of
the field with dimensions).
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solve: Develop and carry out a process for solving a
problem.
V
sort: Separate a set of objects, drawings, ideas, or
numbers according to an attribute (e.g., sort 2-D figures
by the number of sides).
verify: Work out an answer or solution again, usually in
another way. Show evidence.
validate: Check an idea by showing that it works.
visualize: Form a picture in your head of what something
is like. Imagine.
Mathematical Words
A
acute triangle: A triangle in which each angle measures
less than 90°
angle of elevation (angle of inclination): The angle
between the horizontal and the line of sight when looking
up at an object
adjacent side: The side that is part of an acute angle in a
right triangle, but is not the hypotenuse; for example, AB
is the adjacent side in relation to ∠A.
C
angle of elevation
hypotenuse
angle of inclination: See angle of elevation
side adjacent to ∠A
B
algebraic expression: A collection of symbols, including
one or more variables, and possibly numbers and
operation symbols; for example, 3x ⫹ 6, x, 5x, and
21 ⫺ 2w are all algebraic expressions.
algebraic term: Part of an algebraic expression, often
separated from the rest of the expression by an addition
or subtraction symbol; for example, the expression
2x 2 ⫹ 3x ⫹ 4 has three terms: 2x 2, 3x, and 4.
altitude: A line segment that represents the height of a
polygon, drawn from a vertex of the polygon
perpendicular to the opposite side
analytic geometry: Geometry that uses the xy-axes,
algebra, and equations to describe relations and positions
of geometric figures
axis of symmetry: A line that separates a 2-D figure
into two identical parts; if the figure is folded along
this line, one of the identical parts fits exactly on the
other part.
Glossary
A
B
base: The number that is used as a factor in a power;
for example, in the power 53, 5 is the base.
BEDMAS: A made-up word used to recall the order of
operations; BEDMAS stands for Brackets, Exponents,
Division, Multiplication, Addition, Subtraction.
bimedian: The line that joins the midpoints of two
opposite sides in a quadrilateral
bimedians
angle of declination: See angle of depression
angle of depression (angle of declination): The angle
between the horizontal and the line of sight when looking
down at an object
angle of depression
midpoints
binomial: An algebraic expression that contains two
terms (e.g., 3x ⫹ 2)
bisect: To divide into two equal parts
object
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Glossary
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C
contained angle: The angle between two known sides
Cartesian coordinate system: A plane that contains an
x-axis (horizontal) and a y-axis (vertical), which are used
to describe the location of a point
continuous: A set of data that can be broken down into
smaller and smaller parts, which still have meaning
centroid: The centre of an object’s mass; the point at
which an object balances; the centroid is also known as
the centre of gravity.
centroid
cosine: The ratio of the length of the adjacent side to the
length of the hypotenuse for either acute angle in a right
triangle; the abbreviation for cosine is cos.
cosine law: In any acute triangle,
c 2 = a 2 + b 2 - 2 ab cos C
C
b
a
A
chord: A line segment that joins two points on a curve
circle: The set of all the points in a plane that are the
same distance, called the radius (r), from a fixed point,
called the centre; the formula for the area of a circle is
A = pr 2.
circumcentre: The centre of a circle that passes through
all three vertices of a triangle; the circumcentre is the same
distance from all three vertices.
circumference: The boundary of a circle, or the length of
this boundary; the formula for circumference [as in
definition of “circle”] is C = 2pr , where r is the radius,
or C = pd , where d is the diameter.
coefficient: The factor by which a variable is multiplied;
for example, in the term 5x , the coefficient is 5.
5x
coefficient
variable
c
B
counterexample: An example used to prove that a
hypothesis or conjecture is false
curve of best fit: The curve that best describes the
distribution of points in a scatter plot, usually determined
using a process called regression
curve of good fit: A curve that approximates, or is close
to, the distribution of points in a scatter plot
D
data point: An item of factual information derived from
measurement or research; on a graph created on a
Cartesian plane, each data point is represented as a dot at
the location denoted by the coordinates of an ordered
pair, where (x, y) ⫽ (value of the independent variable,
value of the dependent variable).
collinear: A word used to describe three or more points
that lie on the same line
decompose: Break a number or an expression into the
parts that make it up
complementary angles: Two angles whose sum is 90°
degree: For a power with one variable, the exponent of
the variable; for an expression with more than one variable,
the sum of the exponents of the powers of the variables; for
example, x 4, x 3y, and x 2y 2 all are degree 4.
completing the square: A process used to rewrite
a quadratic relation that is in standard form,
y = ax 2 + bx + c, in its equivalent vertex form,
y = a(x - h)2 + k
congruent: Equal in all respects; for example, in two
congruent triangles, the three corresponding pairs of sides
and the three corresponding pairs of angles are equal.
conjecture: A guess or prediction based on limited
evidence
constant: A value in a mathematical expression or
formula that does not change; for example, in the
expression 3x ⫹ 2, 2 is a constant.
530
Glossary
denominator: The number in a fraction that represents
the number of parts in the whole set, or the number
of parts that the whole set has been divided into;
4
for example, in , the denominator is 5.
5
dependent variable: In a relation, the variable whose
values you calculate; the dependent variable is usually
placed in the right column of a table of values and on the
vertical axis in a graph.
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diagonal: In a polygon, a line segment joining two
vertices that are not next to each other (not joined by
one side)
diameter: A line segment that joins two points on a
circle and passes through the centre, or the length of this
line segment
difference of squares: An expression of the form a 2 - b 2,
which involves the subtraction of two squares
dilatation: A transformation that enlarges or reduces a
figure
equivalent systems of linear equations: Two or more
systems of linear equations that have the same solution
expand: To write an expression in extended but
equivalent form; for example, 3(5x + 2) = 15x + 6
exponent: The number that tells how many equal factors
are in a power
exterior angle: The angle that is formed by extending a
side of a convex polygon, or the angle between any
extended side and its adjacent side
exterior angle
direct variation: A relation in which one variable is a
multiple of the other variable
discriminant: The expression b 2 - 4ac in the quadratic
formula
distributive property or law: The property or law
stating that when a sum is multiplied by a number,
each value in the sum is multiplied by the number
separately, and the products are then added (for example,
4 * 17 + 82 = (4 * 7) + (4 * 8))
divisor: A number by which another is divided; for
example, in 18 ⫼ 3 ⫽ 6, 3 is the divisor.
E
elimination strategy: A method of removing a variable
from a system of linear equations by creating an
equivalent system in which the coefficients of one of the
variables are the same or opposites
equation: A mathematical sentence in which the value on
the left side of the equal sign is the same as the value on
the right side of the equal sign; for example, the equation
5n ⫹ 4 ⫽ 39 means that 4 more than the product of 5 and
a number equals 39.
equation of a line: An equation of degree 1, which gives
a straight line when graphed on the Cartesian plane;
an equation of a line can be expressed in several forms:
Ax + By = C, Ax + By + C = 0, and y = mx + b are
the most common. For example, the equations
4x ⫹ 2y ⫽ 8, 4x ⫹ 2y ⫺ 8 ⫽ 0, and y ⫽ ⫺2x ⫹ 4 all
represent the same straight line when graphed.
equilateral triangle: A triangle that has all sides equal
in length
equivalent equations: Equations that have the same
solution
NEL
F
factor: To express a number as the product of two or
more numbers, or express an algebraic expression as the
product of two or more terms
factored form of a quadratic relation: A quadratic
relation that is written in the form y = a(x - r)(x - s)
finite differences: Differences between the y-values in
a table of values in which the x-values increase by the
same amount
first difference: Values that are calculated by subtracting
consecutive y-values in a table of values that has a constant
difference between the x-values
G
greatest common factor (GSF): The greatest factor of
two or more terms
H
hypotenuse: The longest side of a right triangle; the side
that is opposite the right angle
I
independent variable: In a relation, the variable whose
values you choose; the independent variable is usually
placed in the left column of a table of values and on the
horizontal axis in a graph.
Glossary
531
Glossary
dividend: A number being divided; for example, in
18 ⫼ 3 ⫽ 6, 18 is the dividend.
extrapolate: To predict a value by following a pattern
beyond known values
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integers (I): All positive and negative whole numbers,
including zero: ... ⫺3, ⫺2, ⫺1, 0, 1, 2, 3, ....
line of good fit: A straight line that approximates, or is
close to, the distribution of points in a scatter plot
interior angle: The angle that is formed inside each
vertex of a polygon; for example, ^ ABC has three interior
angles: ∠ABC, ∠ BCA, and ∠ CAB.
B
∠ABC
line of symmetry: A line that divides a figure into two
congruent parts, which can be matched by folding the
figure in half
∠BCA
A
∠CAB
C
interpolate: To estimate a value between two known values
intersecting lines: Lines that cross each other and have
exactly one point in common; this point is called the
point of intersection.
line segment: The part of a line between two endpoints
M
maximum value: The greatest value of the dependent
variable in a relation
mean: A measure of central tendency, which is calculated
by dividing the sum of a set of numbers by the number of
numbers in the set
median: A line that is drawn from a vertex of a triangle
to the midpoint of the opposite side
inverse: The reverse of an original statement; for example,
if x = sin u, the inverse is u = sin - 1x.
inverse operations: Operations that undo, or reverse,
each other; for example, addition is the inverse of
subtraction, and multiplication is the inverse of division.
isolating a term or a variable: Performing math operations
(e.g., addition, subtraction, multiplication, or division) to
get a term or a variable by itself on one side of an equation
median
midpoint: The point that divides a line segment into two
equal parts
midsegment: A line segment that connects the midpoints
of two adjacent sides of a polygon
isosceles triangle: A triangle that has two sides equal in
length
midsegment
K
kite: A quadrilateral that has two pairs of equal sides
but no parallel sides
midsegment of a quadrilateral: A line segment that
connects the midpoints of two adjacent sides in a
quadrilateral
minimum value: The least value of the dependent
variable in a relation
L
like terms: Algebraic terms that have the same variables
and exponents, apart from their numerical coefficients
(e.g., 2x 2 and ⫺3x)
linear equation: An equation of the form a x + b = 0,
or an equation that can be rewritten in this form; the
algebraic expression in a linear equation is a polynomial of
degree 1 (e.g., 2x ⫹ 3 ⫽ 6 or y ⫽ 3x ⫺ 5).
linear relation: A relation in which the graph forms a
straight line
line of best fit: A line that best describes the relationship
between two variables in a scatter plot
532
Glossary
monomial: An algebraic expression that has one term
(e.g., 5x 2, 4x y)
N
negative correlation: A correlation in which one variable
in a relationship increases as the other variable decreases,
and vice versa
negative reciprocals: Numbers that multiply to produce
3
4
⫺1; for example, and ⫺ are negative reciprocals
4
3
1
of each other, as are ⫺ and 2.
2
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nonlinear relation: A relation whose graph is not a
straight line
perpendicular bisector: A line that bisects a line segment
and is perpendicular to the line segment
numerator: The number in a fraction that represents the
number of parts of the size given by the denominator of
the fraction
perpendicular lines: Lines that are in the same plane and
intersect at a 90° angle
O
opposite side: The side that is directly across from a
specific acute angle in a right triangle; for example, BC is
the opposite side in relation to ∠ A.
C
hypotenuse
side
opposite
to ∠A
A
B
polynomial: An expression that consists of a sum and/or
difference of monomials
positive correlation: A correlation in which both
variables in a relationship increase or decrease together
power: A numerical expression that shows repeated
multiplication; for example, the power 53 is a shorter way
of writing 5 ⫻ 5 ⫻ 5. A power has a base and an
exponent: the exponent tells the number of equal factors
in the power.
primary trigonometric ratios: The basic ratios of
trigonometry (sine, cosine, and tangent)
proportion: An equation that consists of equivalent ratios
(e.g., 2:3 = 4:6)
orthocentre: The point where the three altitudes of a
triangle intersect
Pythagorean theorem: The conclusion that, in a right
triangle, the square of the length of the longest side is
equal to the sum of the squares of the lengths of the other
two sides
P
Q
parabola: A symmetrical graph of a quadratic relation,
shaped like the letter “U” right-side up or upside down
quadratic equation: An equation that contains at least
one term whose highest degree is 2; for example,
x2 + x - 2 = 0
parallel lines: Lines in the same plane that do not
intersect
parallelogram: A quadrilateral that has equal and parallel
opposite sides; for example, a rhombus, rectangle, and
square are all types of parallelograms.
parameter: A coefficient that can be changed in a
relation; for example, a, b, and c are parameters in
y = ax 2 + bx + c.
partial variation: A relation in which one variable is a
multiple of the other, plus a constant amount
perfect square: A number or term that has two identical
factors
perfect-square trinomial: A trinomial that has two
identical binomial factors; for example, x 2 + 6x + 9 has
the factors (x + 3)(x + 3).
NEL
quadratic formula: A formula for determining the roots
of a quadratic equation of the form ax 2 + bx + c = 0;
the quadratic formula is written using the coefficients and
the constant in the equation:
x =
- b ; 2b 2 - 4ac
2a
quadratic regression: A process that fits the second
degree relation y = ax 2 + bx + c to the data
quadratic relation in standard form: A relation of
the form y = ax 2 + bx + c, where a Z 0; for example,
y = 3x 2 + 4x - 2
quadrilateral: A polygon that has four sides
quotient: The result of dividing one number by another
number; for example, in 12 , 5 = 2.4, 2.4 is the
quotient.
Glossary
533
Glossary
order of operations: Rules that describe the sequence to
use when evaluating an expression:
1. Evaluate within brackets.
2. Calculate exponents and square roots.
3. Divide or multiply from left to right.
4. Add or subtract from left to right.
point of intersection: A point that two lines have in
common
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R
radius ( plural radii): A line segment that goes from the
centre of a circle to its circumference, or the length of this
line segment
rate of change: The change in one variable relative to the
change in another variable
ratio: A comparison of quantities with the same units
3
(e.g., 3:5 or )
5
rational numbers (Q): Numbers that can be expressed as
the quotient of two integers, where the divisor is not 0
real numbers: The set of numbers that corresponds to
each point on the number line shown; fractions, decimals,
integers, and numbers like 12 are all real numbers.
⫺1
0
1
rectangle: A parallelogram that has four square corners
reflection: A transformation in which a 2-D figure is flipped;
each point in the figure flips to the opposite side of the line
of reflection, but stays the same distance from the line
relation: A description of how two variables are connected
rhombus: A parallelogram with four equal sides
right angle: An angle that measures 90°
right triangle: A triangle that contains one 90° angle
rise: The vertical distance between two points
root: A solution; a number that can be substituted for the
variable to make the equation a true statement; for
example, x = 1 is a root of x 2 + x - 2 = 0, since
12 + 1 - 2 = 0 .
rotation: A transformation in which a 2-D figure is
turned about a centre of rotation
run: The horizontal distance between two points
second differences: Values that are calculated by
subtracting consecutive first differences in a table of values
similar triangles: Triangles in which corresponding sides
are proportional and corresponding angles are equal
sine: The ratio of the length of the opposite side to the
length of the hypotenuse for either acute angle in a right
triangle; the abbreviation for sine is sin.
a
b
c
=
=
sine law: In any acute triangle,
sin A
sin B
sin C
C
b
a
A
c
B
slope: A measure, often represented by m, of the
steepness of a line; or the ratio that compares the vertical
and horizontal distances (called the rise and run) between
¢y
rise
=
two points: m =
run
¢x
solution to an equation: The value of a variable that
makes an equation true; for example, in the equation
5n + 4 = 39, the value of n is 7 because 5172 + 4 = 39.
solution to a system of linear equations: The values of
the variables in the system that satisfy all the equations;
for example, (7, 3) is the solution to x + y = 10 and
4x - 2y = 22.
solve for a variable in terms of other variables: The
process of using inverse operations to express one
variable in terms of the other variable(s)
square: A rectangle that has four equal sides
substitution strategy: A method in which a variable in
one expression is replaced with an equivalent expression
from another expression, when the value of the variable is
the same in both
supplementary angles: Two angles whose sum is 180°
S
scale factor: The value of the ratio of corresponding side
lengths in a pair of similar figures
system of linear equations: A set of two or more linear
equations with two or more variables; for example,
x + y = 10 and 4x - 2y = 22
scalene triangle: A triangle that has three sides of
different lengths
T
scatter plot: A graph that attempts to show a relationship
between two variables by means of points plotted on a
coordinate grid. It is also called a scatter diagram.
534
Glossary
table of values: An orderly arrangement of facts,
displayed in a table for easy reference; for example, a table
of values could be an arrangement of numerical values in
vertical and horizontal columns.
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tangent: The ratio of the length of the opposite side to
the length of the adjacent side for either acute angle in a
right triangle; the abbreviation for tangent is tan.
transformation: A change in a figure that results in
a different position, orientation, or size; for example,
a translation, a reflection, a rotation, a compression, a
stretch, and a dilatation are all transformations.
translation: A transformation in which a 2-D figure is
shifted left or right, or up or down; each point in the figure
is shifted the same distance and in the same direction.
transversal: A line that intersects or crosses two or more
lines
trapezoid: A quadrilateral that has one pair of parallel sides
trend: A relationship between two variables, with time as
the independent variable
trigonometry: The branch of mathematics that deals with
the properties of triangles and calculations based on these
properties
vertex (plural vertices): The point of intersection of a
parabola and its axis of symmetry
vertex form: A quadratic relation of the form
y = a(x - h)2 + k, where the vertex is (h, k)
vertical compression: A transformation that decreases all
the y-coordinates of a relation by the same factor
vertical stretch: A transformation that increases all the
y-coordinates of a relation by the same factor
volume: The amount of space that is occupied by an
object
X
x-intercept: The value at which a graph meets the
x-axis; the value of y is zero for all x-intercepts.
Y
y-intercept: The value at which a graph meets the
y-axis; the value of x is zero for all y-intercepts.
Z
V
zero principle: Two opposite integers that, when added,
give a sum of zero (for example, 1-12 + 1+ 12 = 0)
variable: A symbol used to represent an unspecified
number; for example, x and y are variables in the
expression x ⫹ 2y.
NEL
zeros of a relation: The values of x for which a relation
equals zero; the zeros of a relation correspond to the
x-intercepts of its graph.
Glossary
535
Glossary
trinomial: An algebraic expression that contains three
terms (e.g., 2x 2 ⫺ 6x y ⫹ 7)
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Review of Essential Skills
and Knowledge
A–1
Operations with Integers
460
A–2
Operations with Rational Numbers
462
A–3
Exponent Laws
463
A–4
The Pythagorean Theorem
464
A–5
Evaluating Algebraic Expressions and Formulas
466
A–6
Determining the Intercepts of Linear Relations
467
A–7
Graphing Linear Relations
469
A–8
Expanding and Simplifying Algebraic Expressions
471
A–9
Solving Linear Equations Algebraically
472
A–10
First Differences and Rate of Change
473
A–11
Creating Scatter Plots and Lines or Curves of Good Fit
475
A–12
Interpolating and Extrapolating
477
A–13
Transformations of Two-Dimensional Figures
479
A–14
Ratios, Rates, and Proportions
481
A–15
Properties of Triangles and Angle Relationships
483
A–16
Congruent Figures
485
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Appendix A
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A–1 Operations with Integers
Integers are all the counting numbers, their opposites, and zero.
Set of integers: I {…, 3, 2, 1, 0, 1, 2, 3, …}
Addition
To add two integers,
• if the integers have the same sign, then the sum has the same sign:
12 (5) 17
• if the integers have different signs, then the sum takes the sign of the
integer farthest from 0: 18 (5) 13
Subtraction
To subtract one integer from another integer, add the opposite.
15 (8) 15 8
7
Multiplication and Division
To multiply or divide two integers,
• if the two integers have the same sign, then the answer is positive:
6 * 8 = 48, -36 , (-9) = 4
• if the two integers have different signs, then the answer is negative:
- 5 * 9 = - 45, 54 , (-6) = - 9
More Than One Operation
Follow the order of operations.
B
Brackets
E
Exponents
D Division
M Multiplication
f
from left to right
A
S
f
from left to right
Addition
Subtraction
EXAMPLE
a)
b)
c)
d)
-10 + (-12)
-12 - (-7)2
-11 + ( -4) + 12 + (-7) + 18
-6 * 9 , 3
20 + ( -12) , (-3)
e)
(-4 + 12) , (-2)
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Solution
Appendix A
a) - 10 + ( - 12) = - 22
b) - 12 - (- 7)2 = - 12 - 49
= - 12 + (-49)
= - 61
c) - 11 + (- 4) + 12 + ( -7) + 18
= - 22 + 30
= 8
d) - 6 * 9 , 3
= - 54 , 3
= - 18
20 + (- 12) , ( - 3)
e)
( - 4 + 12) , (- 2)
20 + 4
=
8 , ( - 2)
24
=
-4
= -6
Appendix B
Glossary
Practice
1. Evaluate.
4. Evaluate.
a) 6 (3)
b) 12 (13)
c) 17 7
d) 23 9 (4)
e) 24 36 (6)
f ) 32 (10) (12) 18 (14)
2. Which sign would make each statement true:
, , or ?
a) 5 4 3 3 ■ 4 3 1 (2)
b) 4 6 6 8 ■ 3 5 (7) 4
c) 8 6 (4) 5 ■ 5 13 7 (8)
d) 5 13 7 2 ■ 4 5 (3) 5
3. Evaluate.
NEL
5. Evaluate.
a)
b)
c)
d)
e)
f)
-12 - 3
-3 - 2
-18 + 6
(-3)( -4)
(-16 + 4) , 2
8 , (- 8) + 4
-5 + (-3)( - 6)
(-2)2 + ( -3)2
(-2)3(9)
62 , ( -4)
(5)2 - ( -3)2
[( -2)(4)]2
Appendix A: Review of Essential Skills and Knowledge
Index
a) 11 (5)
b) 3(5)(4)
c) 35 (5)
d) 72 (9)
e) 5(9) (3)(7)
f ) 56 [(8)(7)] 49
Answers
a) (-3)2 - ( -2)2
b) ( -5)2 - ( -7) + (-12)
c) -4 + 20 , (- 4)
d) -3(-4) + 82
e) -16 - [( -8) , 2]
f ) 8 , (-4) + 4 , (-2)2
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A–2 Operations with Rational Numbers
Rational numbers are numbers that can be expressed as the quotient
of two integers, where the divisor is not zero.
a
Set of rational numbers: Q = e ` a, b I, b Z 0 f
b
Addition and Subtraction
To add or subtract rational
numbers, determine a common
denominator.
Multiplication
To multiply rational numbers, first
reduce to lowest terms (if possible).
a
c
ac
* =
b
d
bd
Division
To divide by a rational number,
multiply by the reciprocal.
c
a
d
a
ad
, = * =
c
b
d
b
bc
EXAMPLE 1
Simplify.
More Than One Operation
Follow the order of operations.
EXAMPLE 2
3
3
-2
+
5
-2
10
Simplify.
-4
-3
3
*
,
4
5
7
Solution
Solution
3
3
-4
- 15
3
-2
+
=
+
5
-2
10
10
10
10
- 4 - 15 - 3
=
10
- 22
1
=
or - 2
10
5
3
-4
-3
3
-4
7
*
,
= *
*
4
5
7
4
5
-3
1
-1
3
-4
=
*
4
5
*
7
-3
1
=
-1
-7
2
or 1
-5
5
Practice
1. Evaluate.
1
-3
+
4
4
1
-2
b)
2
3
a)
c)
-1
2
*
3
-5
1
3
d) - 4 * a-7 b
6
4
-3
-2
,
3
8
1
1
f ) a- 2 b , a- 3 b
3
2
3
1
-2
c) a b a
ba
b
5
-6
3
-2 2 1 3
d) a
b a
b
3
-2
e) a
e)
2. Evaluate.
-2
-1
1
- a
+
b
5
10
-2
-1
-3 -3
b)
a
b
5
4
4
a)
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Appendix A: Review of Essential Skills and Knowledge
-2
1
5
-1
+
b , a
b
5
-2
-8
2
-4
-3
1
-1
f)
* 5
5
3
5
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A–3 Exponent Laws
exponent
34
an
and
4 factors of 3
34 = (3)(3)(3)(3)
Operations with powers follow a set of rules.
Appendix A
base
34 and a n are called powers.
n factors of a
a n = (a)(a)(a)…(a)
Description
Algebraic Expression
Example
Multiplication
When the bases are the
same, keep the base the
same and add exponents.
(am)(an) = am + n
(54)(5-3) = 54 + (-3)
= 54 - 3
= 51
= 5
Division
When the bases are the same,
keep the base the same and
subtract exponents.
am
= am-n
an
46
= 46 - (-2)
4-2
= 46 + 2
= 48
Power of a Power
Keep the base, and multiply
the exponents.
(am)n = amn
(32)4 = 3(2)(4)
= 38
Appendix B
Rule
EXAMPLE
Simplify and evaluate.
3(37) , (33)2
Solution
Glossary
3(37) , (33)2 = 31 + 7 , 33 * 2
= 38 , 36
= 38 - 6
= 32
= 9
Practice
1. Evaluate to three decimal places, if necessary.
b) 50
c)
32
e)
(-5)3
1 3
f) a b
2
d) - 32
2. Evaluate.
a) 30 + 50
b) 22 + 33
c) 52 - 42
3
98
97
21552
b)
53
NEL
c) (45)(42)3
d)
a) (x 5)(x 3)
b) (m 2)(m 4)(m 3)
c) ( y 5)( y 2)
d) (a b)c
1x 521x 32
e)
x2
x4 3
f) a 3 b
y
5. Simplify.
a) (x 2y 4)(x 3y 2)
b) (2m3)2(3m2)3
15x 222
c)
15x 220
d) (4u3v2)2 (2u2v3)2
Index
1
2
d) a b a b
2
3
5
e) -2 + 24
1 2
1 2
f) a b + a b
2
3
3. Evaluate to an exact answer.
a)
2
4. Simplify.
Answers
a)
42
13221332
13422
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A–4 The Pythagorean Theorem
a
right angle
c
hypotenuse
b
The three sides of a right triangle are related to each other in a unique
way, through an important mathematical relationship called the
Pythagorean theorem.
Every right triangle has a side called the hypotenuse, which is always the
longest side and opposite the right angle. According to the Pythagorean
theorem, the area of the square of the hypotenuse is equal to the sum
of the areas of the squares of the other two sides.
c2
a2
b2
a2
⫹
b2
⫽
c2
EXAMPLE
Determine each indicated side length. Round your answers
to one decimal place, if necessary.
a)
12 cm
5 cm
?
b)
14.0 cm
20.0 cm
?
Solution
a) Let a = 5, b = 12, and c = ?.
a2 + b2 = c 2
52 + 122 = c 2
25 + 144 = c 2
169 = c 2
2169 = c
13 = c
The length is 13 cm.
b) Let a = 14.0, b = ?, and c = 20.0.
a2 + b2 = c 2
14.02 + b 2 = 20.02
196.00 + b 2 = 400.00
b 2 = 400.00 - 196.00
b 2 = 204.00
b = 2204.00
b 14.3
The length is about 14.3 cm.
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Practice
4. Determine the length of the diagonals
right triangle.
a)
Appendix A
1. Write the Pythagorean theorem for each
in each rectangle to one decimal place.
a)
5m
x
6 cm
10 m
8 cm
b)
Appendix B
b)
6 cm
c
13 cm
3 cm
c)
5.2 cm
6 cm
Glossary
c)
5.2 cm
d) 1.2 m
9m
4.8 m
y
5. An isosceles triangle has a hypotenuse that is
15.0 cm long. Determine the length of the two
equal sides.
5m
a
d)
of the shadow is 100.0 m from the top of the
building and 72.0 m from the base of the
building. How tall is the building?
8.5 cm
Answers
3.2 cm
6. An apartment building casts a shadow. The tip
2. Calculate the length of the unknown side
in each triangle in question 1. Round your
answers to one decimal place, if necessary.
100.0 m
3. Determine the value of each unknown measure
NEL
72.0 m
Index
to two decimal places, if necessary.
a) a 2 = 52 + 132
b) 102 = 82 + m 2
c) 262 = b 2 + 122
d) 2.32 + 4.72 = c 2
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A–5 Evaluating Algebraic Expressions
and Formulas
To evaluate algebraic expressions and formulas, substitute the given numbers
for the variables. Then follow the order of operations to calculate the answer.
EXAMPLE 1
EXAMPLE 2
Determine the value of 2x 2 - y
if x = - 2 and y = 3.
The formula for calculating the
volume of a cylinder is V = pr 2h.
Determine the volume of a cylinder
with a radius of 2.5 cm and a height
of 7.5 cm.
Solution
Solution
2x 2 - y = 21-222 - 3
V = pr 2h
13.14212.52217.52
= 13.14216.25217.52
147.2
= 2142 - 3
= 8 - 3
= 5
The volume is about 147.2 cm3.
Practice
1. Determine the value of each expression
3. a) The formula for the area of a triangle is
if x = - 5 and y = - 4.
a) - 4x - 2y
b) - 3x - 2y 2
c) (3x - 4y)2
y
x
d)
y
x
1
2
and y = , calculate the value
2
3
of each expression.
a) x + y
b) x + 2y
c) 3x - 2y
1
1
d)
x - y
2
2
2. If x = -
1
2
A bh. Determine the area of a triangle
if b 13.5 cm and h 12.2 cm.
b) The area of a circle is calculated using the
formula A = pr 2. Determine the area
of a circle with a radius of 4.3 m.
c) The hypotenuse of a right triangle, c, is
calculated using the formula
c = 2a 2 + b 2. Determine the length of
the hypotenuse if a = 6 m and b = 8 m.
d) The volume of a sphere is calculated using
4
the formula V = pr 3. Determine the
3
volume of a sphere with a radius of 10.5 cm.
10.5 cm
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Appendix A
A–6 Determining the Intercepts
of Linear Relations
A linear relation of the general form Ax + By + C = 0 has an x-intercept
and a y-intercept. These are the points where the line Ax + By + C = 0
crosses the x-axis and the y-axis.
EXAMPLE 1
AX BY C 0 FORM
Determine the x- and y-intercepts of the linear relation 2x + y - 6 = 0.
Appendix B
Solution
The x-intercept is where the relation crosses the x-axis. The equation
of the x-axis is y = 0, so substitute y = 0 into 2x + y - 6 = 0.
2x + 0 - 6 = 0
2x = 6
x = 3
To determine the y-intercept, substitute x = 0 into 2x + y - 6 = 0.
2102 + y - 6 = 0
y = 6
Glossary
The x-intercept is at (3, 0), and the y-intercept is at (0, 6).
EXAMPLE 2
ANY FORM
Determine the x- and y-intercepts of the linear relation 3y = 18 - 2x.
Solution
Answers
To determine the x-intercept, substitute y = 0 into the equation.
3y = 18 - 2x
3102 = 18 - 2x
2x = 18
x = 9
To determine the y-intercept, substitute x = 0 into the equation.
3y = 18 - 2x
3y = 18 - 2102
3y = 18
y = 6
NEL
Appendix A: Review of Essential Skills and Knowledge
Index
The x-intercept is at (9, 0), and the y-intercept is at (0, 6).
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A special case is when the linear relation is a horizontal or vertical line.
EXAMPLE 3
Determine either the x- or the y-intercept of each linear relation.
a) 2x = - 14
b) 3y + 48 = 0
Solution
a) 2x = - 14 is a vertical line, so it has no y-intercept.
To determine the x-intercept, solve for x.
2x = - 14
x = -7
The x-intercept is located at (7, 0).
b) 3y + 48 = 0 is a horizontal line, so it has no x-intercept.
To determine the y-intercept, solve for y.
3y + 48 = 0
3y = - 48
y = - 16
The y-intercept is located at (0, 16).
Practice
1. Determine the x- and y-intercepts of each
3. Determine either the x-intercept or y-intercept
linear relation.
a) x + 3y - 3 = 0
b) 2x - y + 14 = 0
c) - x + 2y + 6 = 0
d) 5x + 3y - 15 = 0
e) 10x - 10y + 100 = 0
f ) - 2x + 5y - 15 = 0
2. Determine the x- and y-intercepts of each
of each linear relation.
a) x = 13
b) y = - 6
c) 2x = 14
d) 3x + 30 = 0
e) 4y = - 6
f ) 24 - 3y = 0
4. A ladder resting against a wall is modelled by
linear relation.
a) x = y + 7
b) 3y = 2x + 6
c) y = 4x + 12
d) 3x = 5y - 30
e) 2y - x = 7
f ) 12 = 6x - 5y
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Appendix A: Review of Essential Skills and Knowledge
the linear relation 2y + 9x = 13.5. The x-axis
represents the ground, and the y-axis represents
the wall.
a) Determine the intercepts of the relation.
b) Use the intercepts to graph the relation.
c) What does this tell you about the foot
of the ladder and the top of the ladder?
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A–7 Graphing Linear Relations
Appendix A
The graph of a linear relation (Ax + By + C = 0) is a straight line. The
graph can be drawn if at least two ordered pairs of the relation are known.
This information can be determined in several different ways.
EXAMPLE 1
TABLE OF VALUES
Sketch the graph of 2y = 4x - 2.
Solution
Choose values for x, and substitute these
values into the equation to calculate y.
y
1
2( -1) - 1 = - 3
0
2(0) - 1 = - 1
1
2(1) - 1 = 1
2
2(2) - 1 = 3
EXAMPLE 2
Plot the ordered pairs
as points, and join them
with a straight line.
5
0
⫺5
y
5x
2y ⫽ 4x ⫺ 2
Glossary
x
Appendix B
A table of values can be created. Express
the equation in the form y = mx + b.
2y
4x - 2
=
2
2
y = 2x - 1
⫺5
USING INTERCEPTS
Answers
Sketch the graph of 2x + 4y = 8.
Solution
The intercepts of the line can be calculated.
For the x-intercept, let y = 0.
2x + 4102 = 8
2x = 8
x = 4
The x-intercept is at (4, 0).
NEL
y
5
2x ⫹ 4y ⫽ 8
0
5
Index
For the y-intercept, let x = 0.
2102 + 4y = 8
4y = 8
y = 2
The y-intercept is at (0, 2).
Plot the ordered pairs
as points, and join them
with a straight line.
x
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EXAMPLE 3
USING THE SLOPE AND Y-INTERCEPT
Sketch the graph of y = 3x + 4.
Solution
When the equation is in the form
y = mx + b, m is the slope and
b is the y-intercept.
The line y = 3x + 4 has a slope
of 3 and a y-intercept of 4.
Plot the y-intercept, and use the
rise and the run of the slope to
locate another point on the line.
Join the points with a straight line.
y
7
6
rise ⫽ 3
5
4
3 run ⫽ 1
2 y ⫽ 3x ⫹ 4
x
1
0
⫺1
1 2 3
Practice
1. Express each equation in the form y = mx + b.
4. Graph each equation by determining
the intercepts.
a) x + y = 4
b) x - y = 3
c) 2x + 5y = 10
d) 3x - 4y = 12
a) 3y = 6x + 9
b) 2x - 4y = 8
c) 3x + 6y - 12 = 0
d) 5x = y - 9
2. Graph each equation using a table of values.
a) y = 3x - 1
1
b) y = x + 4
2
c) 2x + 3y = 6
d) y = 4
5. Graph each equation using the slope and
y-intercept.
a) y = 2x + 3
2
b) y = x - 1
3
3
c) y = - x - 2
4
d) 2y = x + 6
3. Determine the x- and y-intercepts of
each equation.
a) x + y = 10
b) 2x + 4y = 16
c) 50 - 10x - y = 0
y
x
d)
+ = 1
2
4
470
6. Graph each equation. Use any strategy.
Appendix A: Review of Essential Skills and Knowledge
a) y = 5x + 2
b) 3x - y = 6
2
c) y = - x + 4
3
d) 4x = 20 - 5y
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A–8 Expanding and Simplifying Algebraic Expressions
Appendix A
An algebraic expression may contain both numbers and letters. The letters are called
variables. An algebraic expression can only be simplified if it contains like terms. Like terms
must have the same variables and the same exponents. For example, 3x 2 and -5x 2 are like
terms and 2xy and 7yz are unlike terms.
A term is the product of a coefficient and a variable. In an algebraic expression, the terms
are separated by addition or subtraction signs. For example, 5x 2 - 3x + 7 has three terms,
where 5 and 3 are the coefficients, 7 is the constant, and x is the variable.
Description
Example
Collecting like terms
2a 3a 5a
Add or subtract the coefficients of the
terms that have the same variables
and exponents.
3a 2b 5a b
3a 5a 2b b
2a b
Distributive property
a(b c) ab ac
Multiply each term of the
binomial by the monomial.
Appendix B
An algebraic expression with one or more terms is called a polynomial. Simple
polynomials have special names: monomial (one term), binomial (two terms),
or trinomial (three terms).
Glossary
4a(2a 3b)
8a2 12ab
Practice
1. Identify the variable and the coefficient
in each expression.
c) 7c 4
b) - 13a
d) - 1.35m
4
y
7
5x
f)
8
e)
2. In each group of terms, which terms are
3. Identify each polynomial as a monomial,
a binomial, or a trinomial.
a) 6x 3 - 5x
d) -yxz 3
3
b) 5x y
e) 5x - 2y
2
c) 7 + 3x - 4x
f ) 3a + 5c - 4b
NEL
5. Expand.
a) 3(2x + 5y - 2)
b) 5x(x 2 - x + y)
c) m 2(3m 2 - 2n)
d) x 5y 3(4x 2y 4 - 2xy 5)
6. Expand and simplify.
a) 3x(x + 2) + 5x(x - 2)
b) -7h(2h + 5) - 4h(5h - 3)
c) 2m 2n(m 3 - n) - 5m 2n(3m 3 + 4n)
d) 3xy3(5x 2y 1) 2xy3(3y 2 7x)
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471
Index
like terms?
a) a, 5x, - 3a, 12a, - 9x
b) c 2, 6c, - c, 13c 2, 1.25c
c) 3xy, 5x 2y, - 3xy, 9x 2y, 12x 2y
d) x 2, y 2, 2xy, - y 2, - x 2, – 4xy
a) 3x + 2y - 5x - 7y
b) 5x 2 - 4x 3 + 6x 2
c) (4x - 5y) - (6x + 3y) - (7x + 2y)
d) m 2n + p - (2p + 3m 2n)
Answers
a) 5x 3
4. Simplify.
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A–9 Solving Linear Equations Algebraically
Any mathematical sentence stating that two quantities are equal is called
an equation; for example, 5x + 6 = 2(x + 5) + 5.
A solution to an equation is a number that makes the left side equal to the
right side. The solution to 5x + 6 = 2(x + 5) + 5 is x = 3. When x is
replaced with 3, both sides of the equation result in 21.
To solve a linear equation, use inverse operations. When necessary,
eliminate fractions by multiplying each term in the equation by the lowest
common denominator. Eliminate brackets by using the distributive
property. Then isolate the variable. A linear equation has only one solution.
EXAMPLE 1
EXAMPLE 2
Solve.
- 3(x + 2) - 3x = 4(2 - 5x)
Solve.
Solution
Solution
- 31x + 22 - 3x = 412 - 5x2
- 3x - 6 - 3x = 8 - 20x
- 3x - 3x + 20x = 8 + 6
14x = 14
14
x =
14
x = 1
y - 2
y - 7
=
3
4
y - 2
y - 7
b = 12a
b
12 a
3
4
41y - 72 = 31y - 22
4y - 28 = 3y - 6
4y - 3y = - 6 + 28
y = 22
y - 2
y - 7
=
3
4
Practice
1. Solve.
3. Determine the solution to each equation.
a) 6x - 8 = 4x + 10
b) 2x + 7.8 = 9.4
c) 13 = 5m - 2
d) 13.5 - 2m = 5m + 41.5
e) 8(y - 1) = 4(y + 4)
f ) 4(5 - r) = 3(2r - 1)
x
= 20
5
2
b)
x = 8
5
3
c) 4 = m + 3
2
a)
5
y = 3 + 12
7
2
1
e) 3y - =
2
3
m
m
f) 4 = 5 +
3
2
d)
2. a) A triangle has an area of 15 cm2 and a base
4. A total of 209 tickets for a concert were sold.
of 5 cm. What is the height of the triangle?
b) A rectangular lot has a perimeter of 58 m
and is 13 m wide. How long is the lot?
There were 23 more student tickets sold than
twice the number of adult tickets. How many
of each type of ticket were sold?
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A–10 First Differences and Rate of Change
Appendix A
When the values of the independent variable x increase by the same
amount throughout a table of values, the differences between the successive
values of the dependent variable y form a table of first differences called a
difference table.
Appendix B
A difference table represents a linear relation if
• the first differences are the same in every row
• a single straight line can be drawn through all the points when they are
plotted on a grid
• the ratio of the difference between the values of x to the corresponding
difference between the values of y is the same for all pairs of points
in the table
Glossary
This ratio represents the rate of change between the variables and
is equivalent to the slope of the line.
¢y
y2 - y1
Rate of change slope =
x2 - x1
¢x
A table of values represents a nonlinear relation if
• the first differences vary; that is, they are not the same
for every row in the difference table
• a single smooth curve, not a straight line, can be drawn through
the points when they are plotted on a grid
EXAMPLE
This table shows the cost to rent a cement mixer for different lengths of time.
0
1
2
3
4
Cost ($)
45
60
75
90
105
Answers
Time (h)
a) Is the relation linear or nonlinear?
b) Graph the data.
c) Calculate the slope. What does the slope represent in this situation?
Solution
Cost ($)
0
45
1
60
2
75
3
90
4
105
First Difference
60 45 15
75 60 15
90 75 15
105 90 15
120
100
80
60
40
20
The first differences are constant, so the relation is linear.
NEL
140
b)
Index
Time (h)
Cost ($)
a)
Cost to Rent a Cement Mixer
y
0
x
1
2
3 4
Time (h)
5
6
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65 - 40
1 - 0
15
=
1
= 15
c) Slope =
The slope represents the rate of change. In this situation, the rate
of change is the rate at which the cost changes as time changes.
The cost increases by $15 for each additional hour.
Practice
1. For each set of data,
i) create a first differences table
ii) determine whether the relation is linear or nonlinear
iii) graph the relationship
a) speed of a falling ball
Time from Release (s)
0
1
2
3
4
5
6
7
8
Speed (m/s)
0.0
9.8
19.6
29.4
39.2
49.0
58.8
68.6
78.4
b) volumes of various cones with a height of 1 cm
Radius of Base (cm)
1
2
3
4
5
6
7
8
Volume of Cone (cm3)
1.047
4.189
9.425
16.755
26.180
37.699
51.313
67.021
c) cost of renting a car for a day
Distance Driven (km)
Cost ($)
0
10
20
30
40
50
60
70
80
45.00
46.50
48.00
49.50
51.00
52.50
54.00
55.50
57.00
1
2
3
4
5
6
7
5800
4850
3900
2950
2000
1050
100
d) value of a photocopier after several years
Age (years)
0
Value of Photocopier ($) 6750
e) population of a town
Year
2001
2002
2003
2004
2005
2006
2007
2008
Population
1560
1716
1888
2077
2285
2514
2765
3042
2. Determine the rate of change for each linear relation in question 1.
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Appendix A
A–11 Creating Scatter Plots and Lines
or Curves of Good Fit
A scatter plot is a graph that shows the relationship between two sets of
numeric data. The points in a scatter plot often show a general pattern, or
trend. A line that approximates a trend for the data in a scatter plot is
called a line of best fit.
A line of best fit passes through as many points as possible, with the
remaining points grouped equally above and below the line.
Appendix B
Data that have a positive correlation have a pattern that slopes up and to
the right. Data that have a negative correlation have a pattern that slopes
down and to the right. If the points nearly form a line, then the correlation
is strong. If the points are dispersed but still form a linear pattern, then the
correlation is weak.
A curve that approximates, or is close to, the data is called
a curve of good fit.
EXAMPLE 1
Glossary
a) Make a scatter plot to display the data in the table. Describe the kind
of correlation that the scatter plot shows.
b) Draw the line of best fit.
Long-Term Trends in Average Number of Cigarettes Smoked per Day
by Smokers Aged 15–19
1981
1983
1985
1986
1989
1990
1991
1994
1995
1996
Number per Day
16.0
16.6
15.1
15.4
12.9
13.5
14.8
12.6
11.4
12.2
Answers
Year
b)
Average Number of Cigarettes
Smoked/Day, Ages 15–19
1981
1985
1989
Year
1993
1997
Average Number of Cigarettes
Smoked/Day, Ages 15–19
18
16
14
12
10
8
6
4
2
0
1981
1985
1989
Year
1993
Index
18
16
14
12
10
8
6
4
2
0
Number ofcigarettes smoked/day
a)
Number ofcigarettes smoked/day
Solution
1997
The scatter plot shows a negative correlation.
NEL
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EXAMPLE 2
A professional golfer is taking part in a scientific investigation. Each time
she drives the ball from the tee, a motion sensor records the initial speed
of the ball. The final horizontal distance of the ball from the tee is also
recorded. Here are the results:
Speed (m/s)
10
16
19
22
38
Distance (m)
10
25
47
43
142 182 244 280
Solution
The scatter plot shows that a line of best
fit does not fit the data as well as an
upward-sloping curve does. Therefore,
sketch a curve of good fit.
50
54
Horizontal Distance of a Golf Ball
300
250
200
150
100
50
Distance (m)
Draw the line or curve of good fit.
43
0
10 20 30 40 50 60
Speed (m/s)
Practice
1. For each set of data,
i) create a scatter plot and draw the line of best fit
ii) describe the type of correlation the trend in the data displays
a) population of the Hamilton–Wentworth, Ontario, region
Year
1966
1976
1986
1996
1998
Population
449 116
529 371
557 029
624 360
618 658
b) percent of Canadians with less than Grade 9 education
Year
1976
1981
1986 1991 1996
Percent of the Population
25.4
20.7
17.7
14.3 12.4
2. In an experiment for a physics project, a marble is rolled up a ramp. A motion sensor detects
the speed of the marble at the start of the ramp, and the final height of the marble is recorded.
The motion sensor, however, may not be measuring accurately. Here are the data:
Speed (m/s)
1.2
2.1
2.8
3.3
4.0
4.5
5.1
5.6
Final Height (m) 0.07
0.21
0.38
0.49
0.86
1.02
1.36
1.51
a) Draw a curve of good fit for the data.
b) How accurate are the motion sensor’s measurements? Explain.
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A–12 Interpolating and Extrapolating
Appendix A
A graph can be used to make predictions about values that are not recorded and plotted. When a prediction
involves a point within the range of values, it is called interpolating. When the value of the independent
variable falls outside the range of recorded data, it is called extrapolating. Estimates from a scatter plot
are more reliable if the data show a strong positive or negative correlation.
EXAMPLE
Solution
Winning Times of Men’s 100 m Run
Name (Country)
Williams (Canada)
Time (s)
10.8
1932
Tolan (U.S.)
10.3
12
1936
Owens (U.S.)
10.3
10
1948
Dillard (U.S.)
10.3
1952
Remigino (U.S.)
10.4
6
1956
Morrow (U.S.)
10.5
4
0
1960
Hary (Germany)
10.2
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
a) Draw a scatter plot, and determine the line of best fit.
Year
1928
1964
Hayes (U.S.)
10.0
Year
1968
Hines (U.S.)
9.95
1972
Borzov (U.S.S.R.)
10.14
Locate 1940 on the x-axis. Follow the vertical line for
1940 up until it meets the line of best fit at about
10.5 s. For 1944, a reasonable estimate is about 10.4 s.
1976
Crawford (Trinidad)
10.06
1980
Wells (Great Britain)
10.25
1984
Lewis (U.S.)
9.99
1988
Lewis (U.S.)
9.92
12
1992
Christie (Great Britain)
9.96
10
1996
Bailey (Canada)
9.84
Time (s)
100 m Run
y
8
Glossary
x
Answers
b) Extend the x-axis to 2024. Then extend the line
of best fit to the vertical line through 2024.
y
8
6
x
Year
The vertical line for 2024 crosses the line of best fit at about 9.7 s. It would be difficult to predict much
farther into the future, since the winning times cannot continue to decline indefinitely.
Appendix A: Review of Essential Skills and Knowledge
477
Index
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
2012
2016
2020
2024
Time (s)
100 m Run
4
0
NEL
Appendix B
The Summer Olympics were cancelled in 1940 and 1944 because of World War II.
a) Estimate what the men’s 100 m run winning times might have been in these years.
b) Predict what the winning time might be in 2024.
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Practice
1. This scatter plot shows the gold medal throws
3. Explain why values you obtained by
in the men’s discus competition in the Summer
Olympics for 1908 to 1992.
extrapolation are less reliable than values
obtained by interpolation.
Discus Throw
4. A school principal wants to know if there is
y
a relationship between attendance and marks.
You have been hired to collect data and analyze
the results. You start by taking a sample
of 12 students.
64
56
48
40
0
1908
1912
1916
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
Distance (m)
72
x
Days
Absent
0
Average
(%)
93 79 81 87 87 75 77 90 77 72 61 80
Year
4
2
0
6
4
1
3
7
8
4
a) Create a scatter plot. Draw the line
of best fit.
b) What appears to be the average decrease
in marks for an absence of 1 day?
c) Predict the average of a student who
is absent for 6 days.
d) About how many days could a student
likely miss before getting an average
below 50%?
a) Estimate what the winning distance might
have been in 1940 and 1944.
b) Estimate the winning distance in 2020 and
2024.
2. As an object falls freely toward the ground, it
accelerates at a steady rate due to gravity. The
table shows the speed, or velocity, that an object
would reach at 1 s intervals during its fall.
3
5. A series of football punts is studied in an
experiment. The initial speed of the football
and the length of the punt are recorded.
Time from Start (s)
Velocity (m/s)
0
0
1
9.8
Speed (m/s)
10
17
18
21
25
2
19.6
Distance (m)
10
28
32
43
61
3
29.4
4
39.2
5
49.0
Use a curve of good fit to estimate the length of
a punt with an initial speed of 29 m/s, as well as
the initial speed of a punt with a length of 55 m.
a) Graph the data.
b) Determine the velocity of the object at 2.5 s,
3.5 s, and 4.75 s.
c) Estimate the velocity of the object obtained
at 6 s, 9 s, and 10 s.
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A–13 Transformations of Two-Dimensional Figures
Appendix A
A translation slides a figure up, down, right, left, or
diagonally along a straight line. For example, ^ABC has been
translated 1 unit right and 3 units down to create the image
triangle, ^A¿B¿C¿ .
B
A
B
C
A
C
y
8
A
A
6
E E
4
B
C
-6 -4
D
-2
Appendix B
A reflection flips a figure about a line of reflection to create
a mirror image. For example, pentagon ABCDE has been
reflected in the y-axis to create the image pentagon,
A¿B¿C¿D¿E¿. The y-axis is the line of reflection.
2
B
D
0
2
C
4
x
6
Glossary
EXAMPLE 1
Describe the transformations that have been performed on figures 1 and 2.
y
6
1
4
2
1
x
0
2
-2
6
Answers
-6
2
Solution
Figure 1 has been translated 9 units right and 2 units down.
Figure 2 has been reflected in the y-axis.
Index
EXAMPLE 2
^ ABC has vertices at A( - 5, 3), B( -2, 3), and C( -2, 1).
State the coordinates of ^ A¿B¿C¿ , if ^ ABC is
a) translated 8 units right and 3 units up
b) reflected in the x-axis
NEL
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Solution
a) When ^ ABC is translated 8 units right and
3 units up, the x-coordinate of each vertex
increases by 8 and the y-coordinate increases
by 3. The coordinates of the image are A¿(3, 6),
B¿ (6, 6), and C¿ (6, 4).
b) When ^ ABC is reflected in the x-axis, the
x-coordinates remain the same but the signs of
the y-coordinates change. The coordinates of the
image are A¿(-5, -3), B¿( -2, -3), and
C¿(- 2, - 1) .
y
8
A (3, 6)
6
B (6, 6)
A(–5, 3)
A(–5, 3) B(–2, 3)
C (6, 4)
x
C(–2, 1)
-8 -6 -4
-2
2
-2
4
6
B(–2, 3) 4
2
x
C(–2, 1)
C (–2, –1)
2
0
y
6
0
2
-2
A (–5, –3) B (–2, –3)
8
Practice
1. In what ways does the image differ from the
4. State the translation that was used
to create each image.
original when each transformation is performed?
a) translation
b) reflection
2. Identify the transformation.
a)
A
B
b) A
y
6
4
B
c)
2
x
D
C
CA
B
-4
C
-2
B
C
5. The coordinates of ^JKL are J(-4, 5),
K( -4, 2), and L(-1, 2). State the coordinates
of ^J¿K ¿L¿ , if ^JKL is
a) translated 2 units right, 5 units down
b) reflected in the x-axis
c) reflected in the y-axis
3. State the reflections that could be
performed on the flag to create each image.
a)
c)
b)
b)
a)
D
A
-2
0
d)
6. Describe how the figure was reflected to create
each image.
a)
y
b)
4
y
4
b)
x
-2 a)
-4
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0
2
-2
0
-2
x
2
-4
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A–14 Ratios, Rates, and Proportions
Appendix A
A ratio compares two or more quantities that are measured in the same
units. Because the units are the same, they are not included in the ratio.
3
A ratio can be written in three different forms: , 3 : 4, and 3 to 4.
4
A rate compares different quantities, so the units must be included.
Both rates and ratios can be simplified by dividing all the terms
by the greatest common factor.
Appendix B
When two ratios are equivalent, they can be used to write
a
c
a proportion: .
b
d
When one of the terms in a proportion is unknown, it can be found by
• using a multiplication or division relationship between the terms
in the numerators and denominators (inspection)
• using inverse operations to solve the equation (algebra)
EXAMPLE 2
In a class of 16 girls and 12 boys,
what is the ratio of boys to girls?
A bouquet of 25 flowers costs $20.
What is the cost per flower?
Solution
Solution
Boys to girls 12 : 16
12 16
:
4 4
3:4
Cost/flower = $20 , 25
$0.80/flower
Glossary
EXAMPLE 1
Answers
EXAMPLE 3
Determine x.
4
14
=
x
21
Solution
NEL
Solving Using Algebra
14
4
21
x
=
is the same as =
x
21
4
14
21
x
=
* 4
4 *
4
14
84
x =
14
x = 6
Appendix A: Review of Essential Skills and Knowledge
Index
Solving by Inspection
4
14
=
x
21
2
4
=
x
3
4 = 2 * 2, so
x = 2 * 3
x = 6
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Practice
1. Express each situation as a ratio.
6. Write the unknown term(s) in each proportion.
a) 2 : 3 = . : 6
b) 3 : 8 = . : 24
c) 8 : 5 = 16 : .
d) 1 : . : 8 = 3 : 12: e) 2 : 5 : . = 6 : : 9
a) 4 pucks to 5 sticks
b) 5 markers to 3 pens
c) 6 boys to 1 girl
d) 7 rabbits to 3 puppies
e) 4 elephants to 7 lions
2. Write each pair of quantities as a ratio, comparing
7. Solve each proportion. Evaluate your answers
to two decimal places, if necessary.
8
x
a)
=
14
40
36
10
b)
=
x
3
5
x
c)
=
7
95
55
13
d)
=
x
42
25
x
e)
=
12
65
18
x
f)
=
24
84
40
21
g)
=
x
35
152
x
h)
=
240
6
lesser to greater. Express the ratio in lowest terms.
a) 10 cm, 8 cm
b) 5 km, 3000 m
c) 30 s, 1.5 min
d) 400 g, 3 kg
e) 5 L, 2500 mL
f ) 40 weeks, 1 year
3. The scale for a scale diagram is 1 cm represents
2 m. Calculate the actual length for each length
on the scale diagram.
a) 2 cm
b) 9 cm
c) 12.25 cm
d) 24.2 cm
e) 13.5 cm
4. Express each situation as a rate.
a) Your heart beats 40 times in 30 s.
b) Your aunt drove 120 km in 1.5 h.
c) You bought 3.5 kg of ground beef
for $17.33.
d) You typed 360 words in 8 min.
5. Solve each proportion.
2
x
=
5
20
4
36
b)
=
x
7
9
24
c)
=
x
12
5
25
d)
=
x
2
a)
482
15
9
=
x
20
x
64
f)
=
15
24
16
20
g)
=
x
65
6
x
h)
=
7
21
8. The ratio of boys to girls at Highland Secondary
School is 3 to 4. If there are 435 boys at the
school, determine the total number of students
at the school.
9. A lawnmower engine requires oil and gas to be
e)
Appendix A: Review of Essential Skills and Knowledge
mixed in the ratio of 2 to 5. If John has 3.5 L
of gas in his can, how much oil should he add?
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A–15 Properties of Triangles and Angle Relationships
Complementary Angles
a + b = 90°
Appendix A
There are several special angle relationships.
Vertically Opposite Angles
a = b
c = d
Isosceles Triangle
∠ A = ∠C
AB = BC
B
a
Appendix B
c
d
b
b
a
A
Supplementary Angles
a + b = 180°
C
Sum of the Angles in a
Triangle
∠A + ∠B + ∠C = 180°
Exterior Angle of a Triangle
a + b = c
Glossary
A
b
a
b
a
B
c
C
Transversal and Parallel Lines
Alternate angles are equal.
c = f,d = g
Corresponding angles are equal.
b = f , a = e, d = h , c = g
NEL
f
c
Index
g
d
Co-interior angles are
supplementary.
d + f = 180°, c + e = 180°
b
a
c
Answers
The longest side is across from
the greatest angle. The shortest
side is across from the least
angle. In this triangle,
∠ A 7 ∠ B 7 ∠C , so
BC 7 AC 7 AB
d
f
e
g
f
d
h
c
e
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Practice
1. Determine the values of x, y, and z in each diagram.
a)
e)
y
y
80°
40°
z
x
z
x
60°
110°
f)
b)
50°
76°
50°
x
z
y
y
x
c)
z
g)
20°
x
x
y
z
y
120°
x
z
d)
50°
z
h)
y
y
110°
x
x
60°
z
60°
2. For each triangle, list the sides from longest to shortest, and the angles from greatest to least.
What do you notice?
a)
b) L
R
N
T
484
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M
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A–16 Congruent Figures
Appendix A
Two figures are congruent if they are the same size and shape. This
relationship is only true when all corresponding angles and all corresponding
sides are equal. The symbol for congruency is .
If ^ ABC ^ LMN , then the following is true.
A
Equal sides:
AB = LM
BC = MN
CA = NL
L
C M
Appendix B
B
Equal angles:
∠ABC = ∠ LMN
∠ BCA = ∠MNL
∠ CAB = ∠ NLM
N
EXAMPLE
Are quadrilaterals AMPJ and RWZE congruent? Give reasons for your answer.
61 cm
92°
70°
A
J
50 cm
48 cm
M
81°
46 cm
P
Glossary
117°
50 cm Z
81°
92°
46 cm
117°
61 cm
W
70°
48 cm
R
E
Solution
Answers
Use the diagram to determine whether the corresponding sides and angles
are equal.
Corresponding angles:
Corresponding sides:
∠ A = ∠ R, ∠J = ∠ E,
JA = ER, PM = WZ
∠ M = ∠ W, ∠ P = ∠ Z
AM = RW, JP = EZ
Quadrilateral AMPJ Quadrilateral RWZE
Practice
1. Draw two congruent rectangles on a coordinate
grid. Explain how you know that your rectangles
are congruent.
3. These two quadrilaterals are congruent.
Determine the values of x, y, and z.
R
2. Quadrilaterals JKLM and SPQR are congruent.
List all the pairs of equal sides and angles.
J
K
P
M
V
D
y 10
60°
K
R
L
M
7 cm
3x
S
Index
S
15 cm
B
45° W
12 z
Q
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Review of Technical Skills
Part 1 Using the TI–83/84 Graphing Calculator
B–1
Preparing the Calculator
487
B–2
Entering and Graphing a Relation
488
B–3
Evaluating a Relation
488
B–4
Changing the Window Settings
489
B–5
Using the Split Screen
489
B–6
Using the Table Feature
489
B–7
Making a Table of Differences
490
B–8
Determining the Zeros of a Relation
491
B–9
Determining the Maximum or Minimum Value of a Relation
491
B–10
Creating a Scatter Plot and Determining a Line or Curve
of Best Fit Using Regression
492
B–11
Determining the Points of Intersection of Two Relations
493
B–12
Evaluating Trigonometric Ratios and Determining Angles
494
B–13
Evaluating Powers and Roots
494
Part 2 Using The Geometer’s Sketchpad
B–14
Defining the Tool Buttons and Sketchpad Terminology
494
B–15
Selections in the Construct Menu
495
B–16
Graphing a Relation on a Cartesian Coordinate System
496
B–17
Creating and Animating a Parameter
497
B–18
Placing Points on a Cartesian Coordinate System Using
Plot Points
499
Placing Points on a Cartesian Coordinate System Using
the Point Tool
499
B–20
Determining the Coordinates of a Point
499
B–21
Constructing a Line, Segment, or Ray through a Given Point
500
B–22
Constructing and Labelling a Point on a Line, Segment, or Ray
500
B–23
Determining the Slope and Equation of a Line
501
B–24
Moving a Line
501
B–25
Constructing a Triangle and Labelling the Vertices
502
B–26
Measuring the Interior Angles of a Triangle
503
B–27
Constructing and Measuring an Exterior Angle of a Triangle
503
B–28
Determining the Sum of the Interior Angles of a Triangle
504
B–29
Measuring the Length of a Line Segment
505
B–30
Constructing the Midpoint of a Line Segment
506
B–31
Constructing the Perpendicular Bisector of a Line Segment
507
B–19
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B–32
Constructing the Bisector of an Angle
507
B–33
Measuring the Area of a Polygon
508
B–34
Constructing a Circle
509
Part 3 Using a Spreadsheet
B–35
510
Introduction to a Spreadsheet
Part 4 Using Fathom
B–36
Creating a Scatter Plot and Determining the Equation
of a Line or Curve of Good Fit
512
Appendix B
Part 5 Using the TI–nspire CAS and TI-nspire Handhelds
Beginning a New Document
513
B–38
Entering and Graphing a Relation
514
B–39
Evaluating a Relation
516
B–40
Changing the Window Settings
517
B–41
Using the Split Screen
518
B–42
Using the Function Table Feature
519
B–43
Making a Table of Differences
520
B–44
Determining the Zeros of a Relation
521
B–45
Determining the Maximum or Minimum Value of a Relation
522
B–46
Creating a Scatter Plot and Determining a Line or Curve
of Best Fit Using Regression
522
B–47
Determining the Points of Intersection of Two Relations
525
B–48
Evaluating Trigonometric Ratios and Determining Angles
526
B–49
Evaluating Powers and Roots
527
Glossary
B–37
Answers
PART 1 USING THE TI-83/84 GRAPHING
CALCULATOR
B–1
Preparing the Calculator
Before you graph a relation, be sure to clear any information left on the calculator
from the last time it was used. You should always do the following:
Index
1. Clear all data in the lists.
Press 2nd
4
ENTER
4
ENTER
1
.
2. Turn off all stat plots.
Press 2nd
Y=
.
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3. Clear all equations in the equation editor.
Y= , and then press
Press
CLEAR
for each equation.
4. Set the window so that the axes range from ⴚ10 to 10.
ZOOM
Press
B–2
. Press WINDOW to verify.
6
Entering and Graphing a Relation
1. Enter the equation of the relation in the equation editor.
To graph y ⫽ 2x ⫹ 8, press
GRAPH
Y=
2
X, T, U, n
1
8
. The graph will be displayed as shown.
2. Enter all linear equations in the form y ⫽ mx ⫹ b.
If m or b are fractions, enter them between brackets. For example, enter
2
7
2x ⫹ 3y ⫽ 7 in the form y = - x + , as shown. Make sure you use the
3
3
(2) key when entering negative numbers.
3. Press
GRAPH
to show the graph.
4. Press
TRACE
to determine the coordinates of any point on the graph.
Use
and
to cursor along the graph.
Press
ZOOM
ENTER
8
TRACE
to trace using integer
intervals. If you are working with several graphs at the same time,
use
B–3
and
(the up and down arrow keys) to scroll between graphs.
Evaluating a Relation
1. Enter the relation in the equation editor.
To enter y ⫽ 2x 2 ⫹ x ⫺ 3, press
X, T, U, n
2
3
Y=
2
X, T, U, n
x2
1
.
2. Use the value operation to evaluate the relation.
To determine the value of the relation at x ⫽ ⫺1, press 2nd
ENTER
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, enter (2)
1
at the cursor, and then press
TRACE
ENTER
.
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Changing the Window Settings
The window settings can be changed to show a graph on a different part
of the xy-axes.
1. Enter the relation in the equation editor.
For example, enter y ⫽ x 2 ⫺ 3x ⫹ 4 in the equation editor.
2. Use the Window function to set the boundaries for each axis.
To display the relation when - 2 … x … 5 and 0 … y … 14, press
1
B–5
ENTER
ENTER
, then
0
, then
1
ENTER
ENTER
3. Press
2
GRAPH
ENTER
ENTER
5
, then
, and finally
4
1
, then
, then
Appendix B
(2)
WINDOW
ENTER
1
.
to show the relation.
Using the Split Screen
Glossary
1. The split screen can be used to see a graph and the equation editor
at the same time.
Press
MODE
and cursor to Horiz. Press
then press 2nd
MODE
ENTER
to select this, and
to return to the home screen. Enter y ⫽ x 2 in
Y1 of the equation editor, and then press
GRAPH
.
2. The split screen can also be used to see a graph and a table at the same time.
MODE
ENTER
Answers
Press
, and move the cursor to G–T (Graph-Table). Press
to select this, and then press
GRAPH
. It is possible to view
the table with different increments. To find out how, see steps 2 and 3 in
Appendix B-6.
B–6
Using the Table Feature
A relation can be displayed in a table of values.
Index
1. Enter the relation in the equation editor.
To enter y ⫽ ⫺0.1x3 ⫹ 2x ⫹ 3, press
X, T, U, n
^
3
1
Y=
2
(2)
X, T, U, n
1
.
1
3
.
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2. Set the start point and step size for the table.
Press 2nd
WINDOW . The cursor is beside TblStart⫽. To start at
x ⫽ ⫺5, press (2)
ENTER
5
. The cursor is now beside
⌬Tblⴝ (⌬, the Greek capital letter delta, stands for change in). To increase
ENTER
1
the x-value by 1s, press
GRAPH
3. To view the table, press 2nd
Use
and
.
.
to move up and down the table. Notice that you can look
at greater or lesser x-values than those in the original range.
B–7
Making a Table of Differences
To make a table with the first and second differences for a relation, use the STAT lists.
1. Press STAT
1
, and enter the x-values into L1.
For y ⫽ 3x 2 ⫺ 4x ⫹ 1, use x-values from ⫺2 to 4. Input each number
ENTER
followed by
.
2. Enter the relation.
Scroll right and up to select L2. Enter the relation y ⫽ 3x 2 ⫺ 4x ⫹ 1, using
L1 as the variable x. Press
4
2
3. Press
2nd
ENTER
1
ALPHA
1
3
1
1
ALPHA
2nd
x2
1
1
.
to display the values of the relation in L2.
4. Determine the first differences.
Scroll right and up to select L3. Then press 2nd
STAT . Scroll right to
OPS and press
7
to choose ¢ List(. Enter L2 by pressing
2nd
)
. Press
2
ENTER
to see the first differences
displayed in L3.
5. Determine the second differences.
Scroll right and up to select L4. Repeat step 4, using L3 instead of L2. Press
ENTER
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Determining the Zeros of a Relation
To determine the zeros of a relation, use the zero operation.
1. Start by entering the relation in the equation editor.
For example, enter y ⫽ ⫺(x ⫹ 3)(x ⫺ 5) in the equation editor. Then press
GRAPH
ZOOM
6
.
2
.
2. Access the zero operation.
TRACE
3. Use
and
of the right zero.
ENTER
Press
Appendix B
Press 2nd
to cursor along the curve to any point that is left
to set the left bound.
4. Cursor along the curve to any point that is right of the right zero.
ENTER
Press
to set the right bound.
Glossary
ENTER again to display the coordinates of the zero
(the x-intercept).
5. Press
6. Repeat to determine the coordinates of the left zero.
B–9
Answers
Determining the Maximum or Minimum
Value of a Relation
The least or greatest value can be found using the Minimum operation or
the Maximum operation.
1. Enter and graph the relation.
For example, enter y ⫽ ⫺2x 2 ⫺ 12x ⫹ 30. Graph the relation, and adjust
the window to get a graph like the one shown. This graph opens downward,
so it has a maximum.
2. Use the maximum operation.
2nd
4
TRACE
TRACE
3
Index
Press 2nd
. For parabolas that open upward, press
to use the Minimum operation.
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3. Use
and
to cursor along the curve to any point that is left
of the maximum value.
Press
ENTER
to set the left bound.
4. Cursor along the curve to any point that is right of the maximum value.
Press
ENTER
to set the right bound.
5. Press
ENTER
again to display the coordinates of the maximum value.
B–10 Creating a Scatter Plot and Determining
a Line or Curve of Best Fit Using Regression
This table gives the height of a baseball above ground, from the time it was hit
to the time it touched the ground.
Time (s)
0
1
2
3
4
5
6
Height (m)
2
27
42
48
43
29
5
You can use a graphing calculator to create a scatter plot of the data.
1. Start by entering the data into lists.
ENTER
Press STAT
. Move the cursor over to the first position in L1,
and enter the values for time. Press
ENTER
after each value. Repeat this
for height in L2.
2. Create a scatter plot.
Press 2nd
Y=
and
1
ENTER
. Turn on Plot 1 by making sure
that the cursor is over On, the Type is set to the graph type you prefer, and L1
and L2 appear after Xlist and Ylist.
3. Display the graph.
Press
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ZOOM
9
to activate ZoomStat.
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4. Apply the appropriate regression analysis.
To determine the equation of the line or curve of best fit, press STAT and
5
scroll over to CALC. In this case, press
4
you need a line of best fit, press
Then press 2nd
,
1
1
to Y-VARS. Press
to enable QuadReg. (When
to enable LinReg(ax⫹b).
2nd
2
,
VARS . Scroll over
twice. This action stores the equation of the line or
Appendix B
curve of best fit into Y1 of the equation editor.
5. Display and analyze the results.
Press
ENTER
. In this example, the letters a, b, and c are the coefficients
of the general quadratic equation y ⫽ ax2 ⫹ bx ⫹ c for the curve of best fit.
R 2 is the percent of data variation represented by the model. The equation
is about y ⫽ ⫺4.90x2 ⫹ 29.93x ⫹ 1.98.
Note: For linear regression, if r is not displayed, turn on the diagnostics
ENTER
0
Glossary
function. Press 2nd
, and scroll down to DiagnosticOn. Press
twice. Repeat steps 4 and 5.
6. Plot the curve.
Press
GRAPH
.
Answers
B–11 Determining the Points of Intersection
of Two Relations
1. Enter both relations in the equation editor.
For example, enter y ⫽ 5x ⫹ 4 into Y1 and y ⫽ ⫺2x ⫹ 18 into Y2.
2. Graph both relations.
Press GRAPH
is displayed.
. Adjust the window settings until the point of intersection
3. Use the intersect operation.
TRACE
5
Index
Press 2nd
.
4. Determine a point of intersection.
You will be asked to verify the two curves and enter a guess (optional) for the
point of intersection. Press
ENTER
after each screen appears. The point
of intersection is exactly (2, 14).
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B–12 Evaluating Trigonometric Ratios
and Determining Angles
1. Put the calculator in degree mode.
Press
MODE
ENTER
. Scroll down and across to Degree. Press
.
2. Use the SIN , COS , or TAN key to calculate a trigonometric ratio.
To determine the value of sin 54°, press SIN
ENTER
5
)
4
.
3. Use SINⴚ1, COSⴚ1, or TANⴚ1 to calculate an angle.
To determine the angle whose cosine is 0.6, press 2nd
)
ENTER .
6
COS
.
B–13 Evaluating Powers and Roots
1. Evaluate the power 5.32.
3
x2
ENTER
5
Press 7
.
-2
3. Evaluate the power 8 .
^
5
ENTER
(2)
2
ENTER
.
Press
5
.
.
2. Evaluate the power 7.55.
Press
8
^
.
4. Evaluate the square root of 46.1
Press 2nd
x2
4
6
.
1
)
ENTER
.
PART 2 USING THE GEOMETER’S SKETCHPAD
B–14 Defining the Tool Buttons and Sketchpad
Terminology
Sketches and Dynamic Geometry
The ability to change objects dynamically is the most important feature of The
Geometer’s Sketchpad. Once you have created an object, you can move it, rotate it,
dilate it, reflect it, or hide it. You can also change its label, colour, shade, or line
thickness. No matter what changes you make, The Geometer’s Sketchpad maintains
the mathematical relationships between your object and other objects it is related
to. This is the principle of dynamic geometry and the basis of the power and
usefulness of The Geometer’s Sketchpad.
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Sketchpad Terminology
menu bar
Select.
Construct a point.
Construct a circle.
Construct a line.
Label.
Custom Tool.
Appendix B
toolbar
display area
Select means move the mouse pointer to the desired location and click the mouse
button (left-click for Windows users).
Deselect means select the selection tool and click anywhere in the display area,
away from any figures you have drawn.
Drag means move the mouse pointer to the point or figure you would like to
move. Click on the point or figure and, while holding down the mouse button,
move the point or figure to a new location. Release the mouse button when the
point or figure is in the desired position. This can also be done to text and labels.
B–15 Selections in the Construct Menu
Command
What It Constructs
What You Must Select
Point On Object
a point on the selected object(s)
one or more segments, rays, lines, or circles
Intersection
a point where two objects intersect
two straight objects, two circles, or a
straight object and a circle
Midpoint
the midpoint of the segment(s)
one or more segments
Segment/Ray/Line
the segment(s), ray(s), or line(s)
defined by the points
two or more points
Parallel Line
the line(s) through the selected
point(s), parallel to the selected
straight object(s)
one point and one or more straight
objects, or one straight object and one or
more points
Perpendicular Line
the line(s) through the selected
point(s), perpendicular to the selected
straight object(s)
one point and one or more straight
objects, or one straight object and one or
more points
Angle Bisector
the ray that bisects the angle defined
by three points
three points (select the vertex second)
Circle By Centre
ⴙ Point
the circle with the given centre and
passing through the given point
two points (select the centre first)
(continued)
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Command
What It Constructs
What You Must Select
Circle By Centre
ⴙ Radius
the circle with the given centre and
with a radius equal to the length of
the given segment
a point and a segment
Arc On Circle
the arc that extends counterclockwise
from the first point on a circle to the
second point
a circle and two points on the circumference
of the circle
Arc Through
3 Points
the arc that passes through the three
given points
three points
Polygon Interior
the interior of a polygon defined by
using the given points as its vertices
three or more points
Circle Interior
the interior of a circle
one or more circles
Sector Interior
the interior of an arc sector
one or more arcs
Arc Segment
Interior
the interior of an arc segment
one or more arcs
Locus
the locus of an object
one geometric object and one point
constructed to lie on a path
Note: At any point in time, your selections determine which menu commands are
available at that point. When a command is not available, it is greyed out in its
menu. Often this means your current selection is not appropriate for that command.
B–16 Graphing a Relation on a Cartesian
Coordinate System
You can graph relations on a Cartesian coordinate
system in The Geometer’s Sketchpad. For example, use
the following steps to graph the relation y = 2x + 8.
1. Turn on the grid.
From the Graph menu, choose Define
Coordinate System.
2. Enter the relation.
From the Graph menu, choose Plot New
Function. The New Function calculator should
appear. Use either the calculator keypad or the
keyboard to enter 2 * x ⫹ 8.
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3. Graph the relation y = 2x + 8.
Press OK on the calculator keypad. The graph
of y = 2x + 8 should appear on the grid.
Appendix B
4. Adjust the origin and/or the scale.
To adjust the origin, click on the point at the
origin to select it. Then click and drag the origin
as desired.
To adjust the scale, click in blank space to deselect.
Then click on the point at (1, 0) to select it.
Click and drag this point to change the scale.
B–17 Creating and Animating a Parameter
You can control relations in The Geometer’s Sketchpad
using parameters. For example, follow these steps to
graph the relation y = ax 2 using a as the parameter.
1. Turn on the grid.
2. Create the parameter.
From the Graph menu, choose New
Parameter.… Enter the Name of the parameter:
a. Change the Value to 3.0, and leave none as
the Units selected.
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3. Click OK.
The parameter will appear on the screen, in the same
format as a label.
4. Enter the relation.
From the Graph menu, choose Plot New Function.
The New Function calculator will appear. Click on
the parameter. The letter a will appear on the
calculator screen. Then use either the calculator
keypad or the keyboard to enter *x^2. Since the value
of the parameter is 3, the relation plotted is y = 3x 2.
Click OK.
5. Change the parameter.
Try a = 2. Use the Selection Tool to double-click on
the parameter, and enter 2. Try other values
for a.
6. Animate the parameter.
Use the Selection Tool to select only parameter a.
From the Display menu, choose Animate Parameter.
The Motion Controller will appear. Use the
,
,
, and
controls to start, stop,
reverse, and pause. Use the Speed box to change
the speed.
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B–18 Placing Points on a Cartesian Coordinate
System Using Plot Points
Sometimes, you may want to plot points without
graphing a relation. For example, suppose that you
want to plot (2, 1), (3, 5), and (–2, 0).
1. Turn on the grid.
2. Enter the coordinates of the points.
Appendix B
From the Graph menu, select Plot Points ….
For each point you want to plot, enter the
x-coordinate followed by the y-coordinate. Use
the Tab key on your keyboard to move from one
coordinate entry space to the next. Select Plot
when you have entered both coordinates of
a point. You can continue entering coordinates
until you click on Done.
B–19 Placing Points on a Cartesian Coordinate
System Using the Point Tool
You can also use the Point Tool to plot points without
graphing a relation. For example, suppose that you
want to plot (4, 2).
1. Turn on the grid.
2. Select the Point Tool.
The selection arrow will now look like a dot. This
indicates that when you click on the grid, a point
will be placed at the location you clicked.
B–20 Determining the Coordinates of a Point
1. Turn on the grid.
2. Plot some points on the grid.
3. Use the Selection Tool to select a point.
4. From the Measure menu, select Coordinates.
The coordinates of the selected point(s) will be
displayed.
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B–21 Constructing a Line, Segment, or Ray
through a Given Point
1. Turn on the grid.
2. Plot the point you want the line (or segment or
ray) to pass through.
3. Plot a second point anywhere on the grid.
4. Shift-click to make sure that both points
are selected.
If you are constructing a ray, make sure that
the point where the ray begins is selected first.
5. From the Construct menu, select Line
(or Segment or Ray).
B–22 Constructing and Labelling a Point on
a Line, Segment, or Ray
1. Turn on the grid.
2. Draw a line (or segment or ray).
3. Select the line by clicking on it.
4. From the Construct menu, select
Point On Line.
5. Select the Label Tool. Use it to double-click on
the point you constructed.
A label will appear beside the point, as well as a
Properties box for the point. You can change the
label of the point by changing the contents of
the label entry in the Properties box.
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B–23 Determining the Slope and Equation of a Line
1. Turn on the grid.
2. Draw a line.
3. Use the Selection Tool to select the line.
4. From the Measure menu, select Slope or
Equation.
Appendix B
B–24 Moving a Line
1. Turn on the grid.
2. Draw a line.
3. Copy the line by clicking Copy and then
Paste from the Edit menu.
4. To keep the new line parallel to the
original line, follow the steps below:
Use the Selection Tool to click and hold only
the line. Hold the mouse button down while
you move the mouse. The new line will move
parallel to the original line.
5. To move the new line so that one point stays
in the same position, follow these steps:
Use the Selection Tool to select a point on the
line, other than the point that is staying in the
same position. Hold the mouse button down
while you move the mouse. The line will move
as the mouse moves, but the original point will
stay fixed.
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B–25 Constructing a Triangle and Labelling
the Vertices
1. Open a new sketch.
2. Use the Point Tool to place three points.
If you hold down the Shift key on your keyboard
while you place the points, all the points will
remain selected as you place them.
3. From the Display menu, select Show Labels.
The order in which you select the points
determines the alphabetical order of the labels.
4. From the Construct menu, select Segments.
The three sides of the triangle will be displayed.
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B–26 Measuring the Interior Angles of a Triangle
1. Open a new sketch, and draw a triangle with
the vertex labels displayed.
2. Shift-click to select the vertices that form
an angle.
For example, to measure ∠ ABC, select
vertex A, then vertex B, and then vertex C.
3. From the Measure menu, select Angle.
Appendix B
4. Repeat steps 2 and 3 for each angle.
B–27 Constructing and Measuring an Exterior
Angle of a Triangle
1. Open a new sketch, and draw a triangle
with the vertex labels displayed.
2. Select two vertices. From the Construct
menu, select Ray.
This will extend one side of the triangle to
form an exterior angle.
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3. While the ray is selected, choose Point On Ray
from the Construct menu.
4. Drag the point so that it is outside the triangle.
Display the label for the point.
5. Select the exterior angle.
Select the point, then the vertex for the angle,
and then the final vertex. From the Measure
menu, select Angle.
B–28 Determining the Sum of the Interior Angles
of a Triangle
1. Open a new sketch, and draw a labelled triangle.
Measure all three interior angles.
Shift-click to select all three angle measures.
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2. From the Measure menu, select Calculate.
A New Calculation window will appear.
Appendix B
3. Use the Values pop-up menu to create the
formula for the sum of the selected angles.
Choose an angle from the Values pop-up.
Then enter ⫹ and continue entering addends
until the formula is complete.
Click OK when you are finished.
B–29 Measuring the Length of a Line Segment
1. Open a new sketch, and draw a line
segment.
2. While the line segment is selected, choose
Length from the Measure menu.
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The length will be displayed.
B–30 Constructing the Midpoint of a Line Segment
1. Open a new sketch, and draw a line segment.
2. While the line segment is selected, choose
Midpoint from the Construct menu.
The midpoint will be displayed.
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B–31 Constructing the Perpendicular Bisector
of a Line Segment
1. Open a new sketch, and draw a line
segment.
2. While the line segment is selected, choose
Midpoint from the Construct menu.
3. Select the segment and the midpoint.
4. From the Construct menu, choose
Perpendicular Line.
Appendix B
The perpendicular bisector will be displayed
as a line through the selected point.
B–32 Constructing the Bisector of an Angle
1. Open a new sketch, and place three points
to form an angle.
2. Use the Ray Tool or the Segment Tool to
draw the angle.
3. Select the vertices that form the angle.
4. From the Construct menu, choose
Angle Bisector.
The angle bisector will be displayed as a ray
going out from the angle.
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B–33 Measuring the Area of a Polygon
1. Open a new sketch. Place points for the
vertices of a polygon and their labels.
Make sure that the vertices are placed and selected
in order in either a clockwise or counterclockwise
direction.
2. While the points are selected, use the Construct
Polygon Interior operation in the Construct
menu to form a polygon.
The Geometer’s Sketchpad will name the polygon
depending on the number of vertices you have
selected.
3. While the interior is highlighted, select Area
from the Measure menu.
The area will be displayed.
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B–34 Constructing a Circle
1. Open a new sketch.
2. Use the Circle Tool to select a point
for the centre of the circle.
To change the size of the circle, hold the mouse
button down and drag the point on the
circumference.
Appendix B
3. Turn on the grid.
To construct a circle with a given centre, passing
through a given point, follow either step 4 or
step 5 below.
4. Using the Circle Tool, click on the centre to
select it.
Place the cursor at the origin. Drag the mouse
to the given point on the circle, and click again.
If your point has integer coordinates, select
Snap Points from the Graph menu before you
begin to drag.
5. Using the Point Tool, draw a point for the
centre and another point to be on the
circumference.
Select the centre and then the point. From the
Construct menu, choose Circle By
CenterⴙPoint.
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PART 3 USING A SPREADSHEET
B–35 Introduction to a Spreadsheet
A spreadsheet is a computer program that can be used to create a table of values
(from data or an equation) and then graph the values as a scatter plot. A spreadsheet
can also be used, when necessary, to determine the equation of the line or curve of
best fit using regression.
A spreadsheet consists of cells that are identified by column letter and row number,
such as A2 or B5. A cell can hold a label, a number, or a formula.
The following table of values shows the time and height of a ball that was thrown
into the air.
Time (s)
0
1
2
3
4
5
Height (m)
2.0
22.1
32.4
32.9
23.6
4.5
Note: Different spreadsheets have different commands. Check the instructions for
your spreadsheet to determine the proper commands to use. The instructions below
are for some versions of Microsoft Excel.
Entering Data to Create a Table
To create a spreadsheet for the data above, open the program to create a new
worksheet. Label cell A1 “time (s)” and cell B1 “height (m).”
Enter the initial values. Enter 0 in A2 and 2.0 in B2. Repeat this process to enter
the rest of the time data in column A and the corresponding heights in column B,
in rows 3 to 7.
Creating a Scatter Plot
Use the cursor to highlight the part of the table that you want to graph.
For this example, select columns A and B in rows 1 to 7. Use the
spreadsheet’s graphing command (
for this program)
to graph the data. A scatter plot like this will appear. Notice that axes labels
are included.
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Performing a Regression
Select any point on the scatter plot, and right click. Select Add
Trendline… in the menu that pops up (or go to Chart in the Menu Bar
and select Add Trendline…). The window at the right will appear.
Appendix B
There are several different types of regressions that are available. (Those
that are grey are not applicable to the current data.) Linear is the default.
In this example, Polynomial of Order 2 (quadratic regression) is used.
Select the Options tab, and you will see several properties that might be
useful. The Forecast option allows you to graph the trendline beyond the
data. (This is useful when extrapolating values.) The y-intercept can be
forced to be a particular value if the value is known. To see the model
equation for your regression analysis, select Display equation on chart.
Click OK. A parabola, graphed with the scatter plot, and the equation of
the parabola will appear.
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PART 4 USING FATHOM
B–36 Creating a Scatter Plot and Determining the
Equation of a Line or Curve of Good Fit
1. Create a case table.
Drag a case table from the object shelf, and drop it in the document.
2. Enter the variables and data.
Click on <new>, type a name for the new variable or attribute, and press
Enter on your keyboard. (If necessary, repeat this step to add more attributes.
Pressing Tab instead of Enter moves you to the next column.) When you
name your first attribute, Fathom creates an empty collection to hold your
data (a little empty box). This is where your data are actually stored. Deleting
the collection deletes your data. When you add cases by typing values, the
collection icon fills with gold balls. To enter the data, click in the blank cell
under the attribute name and begin typing values. (Press Tab to move from
cell to cell.)
3. Graph the data.
Drag a new graph from the object shelf at the top of the
Fathom window, and drop it in a blank space in your
document. Drag an attribute from the case table, and drop it
on the prompt below and/or to the left of the appropriate axis
in the graph.
4. Create a relation.
Right-click on the graph, and select Plot Function. Enter the right
side of your relation using a parameter that can be adjusted to fit the
curve to the scatter plot (a was used below). In this case in the
equation, height was used for the dependent variable and x was used
for the independent variable. Time could also have been used as the
independent variable.
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5. Create a slider for the parameter(s) in your equation.
Drag a new slider from the object shelf at the top of the Fathom
window, and drop it in a blank space below your graph. Over V1,
type the letter of the parameter used in your relation in step 4.
Click on the number, and then adjust the value of the slider until
you are satisfied with the fit. This can be done by moving the
cursor over the slider’s number line and clicking and dragging to
adjust, and dragging the slider as needed.
The equation of a curve of good fit is y = - 4.8(x + 0.2)(x - 6.2).
Appendix B
PART 5 USING THE TI-nspire CAS AND
TI-nspire HANDHELDS
B–37 Beginning a New Document
To begin a new document, you should always do the following:
1. Select a New Document.
and scroll to 6: New Document. Press
.
Glossary
Press
Answers
2. Save the previous document.
If you want to save the previous document, select Yes and then
press
. If you do not want to save the previous document,
select No and then press
using either the
. (You can move the cursor by
key or the and keys.) Selecting No
Index
closes all applications that are open in the document.
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3. Select the application.
Select the application you want to use, and then press
.
(You can move the cursor by using the and keys.) Every time
you add a new application, it appears as a numbered page in your
document. To move between these applications, press
and use
either the or key to scroll to the desired application.
B–38 Entering and Graphing a Relation
Enter the equation of the relation into the Graphs & Geometry data entry line.
The handheld will display the graph.
1. Select 2: Graphs & Geometry from the application menu.
2. Graph.
To show the scale on the graph, press
and scroll to
2: View 8: Show Axes End Values, and press
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3. Enter all equations in the form y = mx + b.
7
2
For example, for 2x + 3y = 7, enter - x + in the data
3
3
entry line. Do so by pressing
.
Appendix B
4. Press
to view the graph.
To move the cursor from the data entry line to the graph,
press
twice.
Glossary
5. To determine the coordinates of any point on the graph,
press
.
Answers
Scroll to 5: Trace 1: Graph Trace, and press
. A point will
appear on the graph. Use the and keys to move the cursor along
the graph. Press
to stop the Trace feature.
Index
If you are working with several graphs at the same time, a trace point
will appear on all relations that have a point in the window shown.
Appendix B: Review of Technical Skills
515
B–39 Evaluating a Relation
1. Enter the relation into the data entry line of the Graphs & Geometry
application.
To enter y = 2x 2 + x - 3, press
Press
.
to display the graph.
2. Use the Points & Lines menu to evaluate the relation.
To determine the value of the relation at x = - 1, press
.
Scroll to 6: Points & Lines 2: Point On and press
to place
a point on the relation.
Move the cursor/pencil to any location on the graph. Press
to
plot the point at this location. The point does not have to be located
at the value you really want. Press
of the point until it is flashing. Press
Press
, then hover over the x-value
to select the x-value.
as many times as necessary to delete all the digits
of the x-value, then enter ⫺1.
Press
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to see the value of the relation at x = - 1.
3. Determine when a relation has a specific y-value.
Hover over the value of the y-coordinate of any point to change it to the required
y-value. For example, to determine when the relation is equal to 3, enter 3 for the
y-value.
Appendix B
The relation has a y-value of 3 when x = - 2.
To determine the other x-value when the y-value is 3, move the point (or plot a new
point) closer to this location, and repeat the previous procedure.
Glossary
B–40 Changing the Window Settings
The window settings can be changed to show a graph on a different part
of the x y-axes.
1. Enter the relation y = x 2 - 3x + 4 into the data entry line of the
Graphs & Geometry application.
2. Select the Window feature to set the boundaries of the graph.
Answers
To display the relation when - 2 … x … 5 and 0 … y … 14,
press
, scroll to 4: Window 1: Window Settings, and
press
. Press
, then
,
then
, then
, then
then
, and finally
. The graph will appear as
,
Index
shown on the next page.
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B–41 Using the Split Screen
To see a graph and a table at the same time: press
, then scroll to
5: Page Layout 2: Select Layout 2: Layout 2 (your choice) and press
518
.
Press
followed by
Press
to select the Lists & Spreadsheet application, and press
Press
and scroll to 5: Function Table 1: Switch to Function Table.
Press
to select the choice, then press
Appendix B: Review of Technical Skills
to move the cursor to the empty screen.
.
again to carry out the request.
B–42 Using the Function Table Feature
A relation can be displayed in a table of values.
1. Enter the relation into the data entry line of the Graphs & Geometry
application.
For example, to enter the relation y = - 0.1x 2 + 2x + 3, press
.
2. Add the Lists & Spreadsheet application.
Press
, and scroll to 3: Lists & Spreadsheet application. Press
.
Appendix B
3. View the function table.
Press
, and scroll to 5: Function Table 1: Switch to Function Table.
Press
to select the choice, and then press
again to carry out the request.
Glossary
4. Set the start point and step size for the table.
the table start at x = - 3 and increase in increments of 0.5, press
,
scroll to 5: Function Table 3: Edit Function Table Settings and press
and then adjust the settings as shown. Then
Answers
It is possible to view the table with different increments. For example, to see
to OK and press
,
.
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B–43 Making a Table of Differences
To create a table with the first and second differences for a relation,
add a List & Spreadsheet application.
1. Add the Lists & Spreadsheet application.
Press
, scroll to 3: Lists & Spreadsheet application, then press
.
For the relation y = 3x 2 - 4x + 1, enter x-values from ⫺2 to 4 into
column A.
2. Enter the relation.
Scroll right and up to select the shaded formula cell for column B.
Enter the relation, using
as the variable x. Press
.
3. Press
to display the values of the relation in column B.
4. Determine the first differences.
Scroll right and up to the shaded formula cell for column C. Then press
. Scroll down until you can choose ¢ List(.
Press
, then
Press
to see the
first differences displayed
in column C.
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.
5. Determine the second differences.
Scroll right and up to select the shaded formula cell for column D.
Repeat step 4, using
instead of
. Press
to show
the second differences displayed in column D.
Appendix B
B–44 Determining the Zeros of a Relation
To determine the zeros of a relation, use the Trace feature in the
Graphs & Geometry application.
1. Enter the relation.
Enter y = - (x + 3)(x - 5) into the data entry line
Glossary
of the Graphs & Geometry application. Press
.
2. Access the Trace feature.
Press
press
and scroll to 5: Trace 1: Graph Trace, then
.
Answers
3. Use the and keys to move the cursor along the curve.
A point will appear on the graph. Using the cursor, move the point until
the word zero is displayed. Repeat to determine the second zero.
Index
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B–45 Determining the Maximum or Minimum
Value of a Relation
The least or greatest value can be found using the Trace feature in the
Graphs & Geometry application.
1. Enter y = - 2x 2 - 12x + 30.
Graph the relation and adjust the window as shown. The graph opens
downward, so it has a maximum.
2. Access the Trace feature.
Press
and scroll to 5: Trace 1: Graph Trace, then press
.
3. Use the and keys to move the cursor along the curve.
A point will appear on the graph. Move the point until the word
maximum is displayed. If a graph has a minimum, the point will be
displayed with the word minimum.
B–46 Creating a Scatter Plot and Determining a
Line or Curve of Best Fit Using Regression
This table gives the height of a baseball above the ground, from the time it was hit
to the time it touched the ground.
Time (s)
0
1
2
3
4
5
6
Height (m)
2
27
42
48
43
29
5
Create a scatter plot of the data:
1. Enter the data into the Lists & Spreadsheet application.
Move the cursor to the top cell of column A, and enter the word time. Enter
the values for time starting in row 1. Press
for height in column B.
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after each value. Repeat this
If you need to resize the width of the column, press
and scroll to
1: Actions 2: Resize 1: Resize Column Width. Use the key to make the
column wider. Press
, then
when the column is wide enough.
Appendix B
2. Create a scatter plot.
to add a Graphs & Geometry application.
Press
and scroll to 3: Graph Type 4: Scatter Plot.
Press
Glossary
Press
.
to select time. Press
again. Press
, then use the arrow keys
Answers
To select time for the x-value, press
to move to y.
Repeat this process to select height.
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3. Display the graph.
Press
and scroll to 4: Window 9: Zoom – Data. Press
.
4. Apply the appropriate regression analysis.
To determine the equation of the line or curve of best fit, return to the
List & Spreadsheet application.
Press
. Move the cursor to the first cell in an empty column. Press
and scroll to 4: Statistics 1: Stat Calculations . Select the appropriate
regression. In this case, select 6: Quadratic Regression and press
Use the
key to select time for the X List, then press
to move to the Y List, use the
524
. Press
key to select height for the
Y List, then press
. Continue to press
highlighted. Press
.
Appendix B: Review of Technical Skills
.
until OK is
5. Display and analyze the results.
In this example, the letters a, b, and c are the coefficients of the
general quadratic equation y = ax 2 + bx + c for the curve of
best fit. R 2 is the percent of data variation represented by the
model. The equation is about y = - 4.90x 2 + 29.93x + 1.98.
6. Plot the curve.
Return to the Graphs & Geometry application. Press
.
The graph type is currently set for a scatter plot. The regression
and scroll to 3: Graph Type
1: Function, and then press
. The regression equation is stored in f1(x).
Press
to see what is written in f1(x). Press
Appendix B
equation is a relation. Press
to show the graph.
Glossary
Answers
B–47 Determining the Points of Intersection of
Two Relations
1. Enter both relations into the data entry line of the Graphs & Geometry
application.
For example, enter y = 5x + 4 in f1(x), and then press
Enter y = - 2x + 18 in f2(x), and then press
.
.
2. Graph both relations.
empty area in the window. Hold the click
Index
Adjust the window settings until the point(s) of intersection is
(are) displayed. You can do this by changing the window settings
or by picking up the graph and moving it. Move the cursor to an
key down until a
closed fist appears. Then use the arrow keys to move the graph
until you see the intersection point(s).
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3. Use the intersection feature.
Press
and scroll to 6: Points & Lines 3: Intersection
Point(s), and then press
.
4. Determine a point of intersection
You will need to select the two lines. Move the cursor to one of
the lines. When the line blinks, press
to select the line.
Repeat this process to select the second line.
As you move the cursor near the second line, a point will appear at the
intersection. To make the point permanent and to know its coordinates,
press
. The point of intersection is (2, 14).
Note: If there is more than one point of intersection, all the points of
intersection that are visible in the window will appear.
B–48 Evaluating Trigonometric Ratios and
Determining Angles
1. Put the handheld in degree mode.
526
Press
. Select 8: System Info 1: Document Settings.
Use
to move through the selections. At Angle select Degree
using the
key.
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to select Degree. Continue to press
Then press
2. Use the
3:42 PM
. Press
,
until OK is selected.
and select the calculator application.
, or
key to calculate trigonometric
ratios.
To determine the value of sin 45°, press
. The answer will be exact. To determine the decimal
, then
.
Appendix B
approximation, press
3. Use SINⴚ1, COSⴚ1, or TANⴚ1 to calculate angles.
To determine the angle whose cosine is 0.6, press
.
Glossary
B–49 Evaluating Powers and Roots
Answers
Use the calculator application.
1. Evaluate the power 5.32.
Press
.
2. Evaluate the power 7.55.
Press
.
Press
Index
3. Evaluate the power 8ⴚ2.
.
4. Evaluate the square root of 46.1.
Press
.
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Glossary
Instructional Words
C
calculate: Figure out the number that answers a question.
Compute.
clarify: Make a statement easier to understand. Provide
an example.
classify: Put things into groups according to a rule and
label the groups. Organize into categories.
compare: Look at two or more objects or numbers and
identify how they are the same and how they are different
(e.g., compare the numbers 6.5 and 5.6; compare the size
of the students’ feet; compare two figures).
conclude: Judge or decide after reflection or after
considering data.
construct: Make or build a model. Draw an accurate
geometric figure (e.g., use a ruler and a protractor to
construct an angle).
create: Make your own example.
D
describe: Tell, draw, or write about what something is
or what something looks like. Tell about a process in a
step-by-step way.
determine: Decide with certainty as a result of
calculation, experiment, or exploration.
extend: 1. In patterning, continue the pattern.
2. In problem solving, create a new problem that takes the
idea of the original problem further.
J
justify: Give convincing reasons for a prediction, an
estimate, or a solution. Tell why you think your answer
is correct.
M
measure: Use a tool to describe an object or determine an
amount (e.g., use a ruler to measure the height or distance
around something; use a protractor to measure an angle;
use balance scales to measure mass; use a measuring cup to
measure capacity; use a stopwatch to measure the time in
seconds or minutes).
model: Show or demonstrate an idea using objects
and/or pictures (e.g., model addition of integers using red
and blue counters).
P
predict: Use what you know to work out what is going to
happen (e.g., predict the next number in the pattern 1, 2,
4, 7, … ).
draw: 1. Show something in picture form (e.g., draw a
diagram).
2. Pull or select an object (e.g., draw a card from the deck;
draw a tile from the bag).
R
E
relate: Describe how two or more objects, drawings,
ideas, or numbers are similar.
estimate: Use your knowledge to make a sensible
decision about an amount. Make a reasonable guess
(e.g., estimate how long it takes to cycle from your home
to school; estimate how many leaves are on a tree; what is
your estimate of 3210 ⫹ 789?).
evaluate: 1. Determine if something makes sense. Judge.
2. Calculate the value as a number.
explain: Tell what you did. Show your mathematical
thinking at every stage. Show how you know.
explore: Investigate a problem by questioning,
brainstorming, and trying new ideas.
528
Glossary
reason: Develop ideas and relate them to the purpose of
the task and to each other. Analyze relevant information
to show understanding.
represent: Show information or an idea in a different way
that makes it easier to understand (e.g., draw a graph;
make a model).
S
show (your work): Record all calculations, drawings,
numbers, words, or symbols that make up the solution.
sketch: Make a rough drawing (e.g., sketch a picture of
the field with dimensions).
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solve: Develop and carry out a process for solving a
problem.
V
sort: Separate a set of objects, drawings, ideas, or
numbers according to an attribute (e.g., sort 2-D figures
by the number of sides).
verify: Work out an answer or solution again, usually in
another way. Show evidence.
validate: Check an idea by showing that it works.
visualize: Form a picture in your head of what something
is like. Imagine.
Mathematical Words
A
acute triangle: A triangle in which each angle measures
less than 90°
angle of elevation (angle of inclination): The angle
between the horizontal and the line of sight when looking
up at an object
adjacent side: The side that is part of an acute angle in a
right triangle, but is not the hypotenuse; for example, AB
is the adjacent side in relation to ∠A.
C
angle of elevation
hypotenuse
angle of inclination: See angle of elevation
side adjacent to ∠A
B
algebraic expression: A collection of symbols, including
one or more variables, and possibly numbers and
operation symbols; for example, 3x ⫹ 6, x, 5x, and
21 ⫺ 2w are all algebraic expressions.
algebraic term: Part of an algebraic expression, often
separated from the rest of the expression by an addition
or subtraction symbol; for example, the expression
2x 2 ⫹ 3x ⫹ 4 has three terms: 2x 2, 3x, and 4.
altitude: A line segment that represents the height of a
polygon, drawn from a vertex of the polygon
perpendicular to the opposite side
analytic geometry: Geometry that uses the xy-axes,
algebra, and equations to describe relations and positions
of geometric figures
axis of symmetry: A line that separates a 2-D figure
into two identical parts; if the figure is folded along
this line, one of the identical parts fits exactly on the
other part.
Glossary
A
B
base: The number that is used as a factor in a power;
for example, in the power 53, 5 is the base.
BEDMAS: A made-up word used to recall the order of
operations; BEDMAS stands for Brackets, Exponents,
Division, Multiplication, Addition, Subtraction.
bimedian: The line that joins the midpoints of two
opposite sides in a quadrilateral
bimedians
angle of declination: See angle of depression
angle of depression (angle of declination): The angle
between the horizontal and the line of sight when looking
down at an object
angle of depression
midpoints
binomial: An algebraic expression that contains two
terms (e.g., 3x ⫹ 2)
bisect: To divide into two equal parts
object
NEL
Glossary
529
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C
contained angle: The angle between two known sides
Cartesian coordinate system: A plane that contains an
x-axis (horizontal) and a y-axis (vertical), which are used
to describe the location of a point
continuous: A set of data that can be broken down into
smaller and smaller parts, which still have meaning
centroid: The centre of an object’s mass; the point at
which an object balances; the centroid is also known as
the centre of gravity.
centroid
cosine: The ratio of the length of the adjacent side to the
length of the hypotenuse for either acute angle in a right
triangle; the abbreviation for cosine is cos.
cosine law: In any acute triangle,
c 2 = a 2 + b 2 - 2 ab cos C
C
b
a
A
chord: A line segment that joins two points on a curve
circle: The set of all the points in a plane that are the
same distance, called the radius (r), from a fixed point,
called the centre; the formula for the area of a circle is
A = pr 2.
circumcentre: The centre of a circle that passes through
all three vertices of a triangle; the circumcentre is the same
distance from all three vertices.
circumference: The boundary of a circle, or the length of
this boundary; the formula for circumference [as in
definition of “circle”] is C = 2pr , where r is the radius,
or C = pd , where d is the diameter.
coefficient: The factor by which a variable is multiplied;
for example, in the term 5x , the coefficient is 5.
5x
coefficient
variable
c
B
counterexample: An example used to prove that a
hypothesis or conjecture is false
curve of best fit: The curve that best describes the
distribution of points in a scatter plot, usually determined
using a process called regression
curve of good fit: A curve that approximates, or is close
to, the distribution of points in a scatter plot
D
data point: An item of factual information derived from
measurement or research; on a graph created on a
Cartesian plane, each data point is represented as a dot at
the location denoted by the coordinates of an ordered
pair, where (x, y) ⫽ (value of the independent variable,
value of the dependent variable).
collinear: A word used to describe three or more points
that lie on the same line
decompose: Break a number or an expression into the
parts that make it up
complementary angles: Two angles whose sum is 90°
degree: For a power with one variable, the exponent of
the variable; for an expression with more than one variable,
the sum of the exponents of the powers of the variables; for
example, x 4, x 3y, and x 2y 2 all are degree 4.
completing the square: A process used to rewrite
a quadratic relation that is in standard form,
y = ax 2 + bx + c, in its equivalent vertex form,
y = a(x - h)2 + k
congruent: Equal in all respects; for example, in two
congruent triangles, the three corresponding pairs of sides
and the three corresponding pairs of angles are equal.
conjecture: A guess or prediction based on limited
evidence
constant: A value in a mathematical expression or
formula that does not change; for example, in the
expression 3x ⫹ 2, 2 is a constant.
530
Glossary
denominator: The number in a fraction that represents
the number of parts in the whole set, or the number
of parts that the whole set has been divided into;
4
for example, in , the denominator is 5.
5
dependent variable: In a relation, the variable whose
values you calculate; the dependent variable is usually
placed in the right column of a table of values and on the
vertical axis in a graph.
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diagonal: In a polygon, a line segment joining two
vertices that are not next to each other (not joined by
one side)
diameter: A line segment that joins two points on a
circle and passes through the centre, or the length of this
line segment
difference of squares: An expression of the form a 2 - b 2,
which involves the subtraction of two squares
dilatation: A transformation that enlarges or reduces a
figure
equivalent systems of linear equations: Two or more
systems of linear equations that have the same solution
expand: To write an expression in extended but
equivalent form; for example, 3(5x + 2) = 15x + 6
exponent: The number that tells how many equal factors
are in a power
exterior angle: The angle that is formed by extending a
side of a convex polygon, or the angle between any
extended side and its adjacent side
exterior angle
direct variation: A relation in which one variable is a
multiple of the other variable
discriminant: The expression b 2 - 4ac in the quadratic
formula
distributive property or law: The property or law
stating that when a sum is multiplied by a number,
each value in the sum is multiplied by the number
separately, and the products are then added (for example,
4 * 17 + 82 = (4 * 7) + (4 * 8))
divisor: A number by which another is divided; for
example, in 18 ⫼ 3 ⫽ 6, 3 is the divisor.
E
elimination strategy: A method of removing a variable
from a system of linear equations by creating an
equivalent system in which the coefficients of one of the
variables are the same or opposites
equation: A mathematical sentence in which the value on
the left side of the equal sign is the same as the value on
the right side of the equal sign; for example, the equation
5n ⫹ 4 ⫽ 39 means that 4 more than the product of 5 and
a number equals 39.
equation of a line: An equation of degree 1, which gives
a straight line when graphed on the Cartesian plane;
an equation of a line can be expressed in several forms:
Ax + By = C, Ax + By + C = 0, and y = mx + b are
the most common. For example, the equations
4x ⫹ 2y ⫽ 8, 4x ⫹ 2y ⫺ 8 ⫽ 0, and y ⫽ ⫺2x ⫹ 4 all
represent the same straight line when graphed.
equilateral triangle: A triangle that has all sides equal
in length
equivalent equations: Equations that have the same
solution
NEL
F
factor: To express a number as the product of two or
more numbers, or express an algebraic expression as the
product of two or more terms
factored form of a quadratic relation: A quadratic
relation that is written in the form y = a(x - r)(x - s)
finite differences: Differences between the y-values in
a table of values in which the x-values increase by the
same amount
first difference: Values that are calculated by subtracting
consecutive y-values in a table of values that has a constant
difference between the x-values
G
greatest common factor (GSF): The greatest factor of
two or more terms
H
hypotenuse: The longest side of a right triangle; the side
that is opposite the right angle
I
independent variable: In a relation, the variable whose
values you choose; the independent variable is usually
placed in the left column of a table of values and on the
horizontal axis in a graph.
Glossary
531
Glossary
dividend: A number being divided; for example, in
18 ⫼ 3 ⫽ 6, 18 is the dividend.
extrapolate: To predict a value by following a pattern
beyond known values
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integers (I): All positive and negative whole numbers,
including zero: ... ⫺3, ⫺2, ⫺1, 0, 1, 2, 3, ....
line of good fit: A straight line that approximates, or is
close to, the distribution of points in a scatter plot
interior angle: The angle that is formed inside each
vertex of a polygon; for example, ^ ABC has three interior
angles: ∠ABC, ∠ BCA, and ∠ CAB.
B
∠ABC
line of symmetry: A line that divides a figure into two
congruent parts, which can be matched by folding the
figure in half
∠BCA
A
∠CAB
C
interpolate: To estimate a value between two known values
intersecting lines: Lines that cross each other and have
exactly one point in common; this point is called the
point of intersection.
line segment: The part of a line between two endpoints
M
maximum value: The greatest value of the dependent
variable in a relation
mean: A measure of central tendency, which is calculated
by dividing the sum of a set of numbers by the number of
numbers in the set
median: A line that is drawn from a vertex of a triangle
to the midpoint of the opposite side
inverse: The reverse of an original statement; for example,
if x = sin u, the inverse is u = sin - 1x.
inverse operations: Operations that undo, or reverse,
each other; for example, addition is the inverse of
subtraction, and multiplication is the inverse of division.
isolating a term or a variable: Performing math operations
(e.g., addition, subtraction, multiplication, or division) to
get a term or a variable by itself on one side of an equation
median
midpoint: The point that divides a line segment into two
equal parts
midsegment: A line segment that connects the midpoints
of two adjacent sides of a polygon
isosceles triangle: A triangle that has two sides equal in
length
midsegment
K
kite: A quadrilateral that has two pairs of equal sides
but no parallel sides
midsegment of a quadrilateral: A line segment that
connects the midpoints of two adjacent sides in a
quadrilateral
minimum value: The least value of the dependent
variable in a relation
L
like terms: Algebraic terms that have the same variables
and exponents, apart from their numerical coefficients
(e.g., 2x 2 and ⫺3x)
linear equation: An equation of the form a x + b = 0,
or an equation that can be rewritten in this form; the
algebraic expression in a linear equation is a polynomial of
degree 1 (e.g., 2x ⫹ 3 ⫽ 6 or y ⫽ 3x ⫺ 5).
linear relation: A relation in which the graph forms a
straight line
line of best fit: A line that best describes the relationship
between two variables in a scatter plot
532
Glossary
monomial: An algebraic expression that has one term
(e.g., 5x 2, 4x y)
N
negative correlation: A correlation in which one variable
in a relationship increases as the other variable decreases,
and vice versa
negative reciprocals: Numbers that multiply to produce
3
4
⫺1; for example, and ⫺ are negative reciprocals
4
3
1
of each other, as are ⫺ and 2.
2
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nonlinear relation: A relation whose graph is not a
straight line
perpendicular bisector: A line that bisects a line segment
and is perpendicular to the line segment
numerator: The number in a fraction that represents the
number of parts of the size given by the denominator of
the fraction
perpendicular lines: Lines that are in the same plane and
intersect at a 90° angle
O
opposite side: The side that is directly across from a
specific acute angle in a right triangle; for example, BC is
the opposite side in relation to ∠ A.
C
hypotenuse
side
opposite
to ∠A
A
B
polynomial: An expression that consists of a sum and/or
difference of monomials
positive correlation: A correlation in which both
variables in a relationship increase or decrease together
power: A numerical expression that shows repeated
multiplication; for example, the power 53 is a shorter way
of writing 5 ⫻ 5 ⫻ 5. A power has a base and an
exponent: the exponent tells the number of equal factors
in the power.
primary trigonometric ratios: The basic ratios of
trigonometry (sine, cosine, and tangent)
proportion: An equation that consists of equivalent ratios
(e.g., 2:3 = 4:6)
orthocentre: The point where the three altitudes of a
triangle intersect
Pythagorean theorem: The conclusion that, in a right
triangle, the square of the length of the longest side is
equal to the sum of the squares of the lengths of the other
two sides
P
Q
parabola: A symmetrical graph of a quadratic relation,
shaped like the letter “U” right-side up or upside down
quadratic equation: An equation that contains at least
one term whose highest degree is 2; for example,
x2 + x - 2 = 0
parallel lines: Lines in the same plane that do not
intersect
parallelogram: A quadrilateral that has equal and parallel
opposite sides; for example, a rhombus, rectangle, and
square are all types of parallelograms.
parameter: A coefficient that can be changed in a
relation; for example, a, b, and c are parameters in
y = ax 2 + bx + c.
partial variation: A relation in which one variable is a
multiple of the other, plus a constant amount
perfect square: A number or term that has two identical
factors
perfect-square trinomial: A trinomial that has two
identical binomial factors; for example, x 2 + 6x + 9 has
the factors (x + 3)(x + 3).
NEL
quadratic formula: A formula for determining the roots
of a quadratic equation of the form ax 2 + bx + c = 0;
the quadratic formula is written using the coefficients and
the constant in the equation:
x =
- b ; 2b 2 - 4ac
2a
quadratic regression: A process that fits the second
degree relation y = ax 2 + bx + c to the data
quadratic relation in standard form: A relation of
the form y = ax 2 + bx + c, where a Z 0; for example,
y = 3x 2 + 4x - 2
quadrilateral: A polygon that has four sides
quotient: The result of dividing one number by another
number; for example, in 12 , 5 = 2.4, 2.4 is the
quotient.
Glossary
533
Glossary
order of operations: Rules that describe the sequence to
use when evaluating an expression:
1. Evaluate within brackets.
2. Calculate exponents and square roots.
3. Divide or multiply from left to right.
4. Add or subtract from left to right.
point of intersection: A point that two lines have in
common
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R
radius ( plural radii): A line segment that goes from the
centre of a circle to its circumference, or the length of this
line segment
rate of change: The change in one variable relative to the
change in another variable
ratio: A comparison of quantities with the same units
3
(e.g., 3:5 or )
5
rational numbers (Q): Numbers that can be expressed as
the quotient of two integers, where the divisor is not 0
real numbers: The set of numbers that corresponds to
each point on the number line shown; fractions, decimals,
integers, and numbers like 12 are all real numbers.
⫺1
0
1
rectangle: A parallelogram that has four square corners
reflection: A transformation in which a 2-D figure is flipped;
each point in the figure flips to the opposite side of the line
of reflection, but stays the same distance from the line
relation: A description of how two variables are connected
rhombus: A parallelogram with four equal sides
right angle: An angle that measures 90°
right triangle: A triangle that contains one 90° angle
rise: The vertical distance between two points
root: A solution; a number that can be substituted for the
variable to make the equation a true statement; for
example, x = 1 is a root of x 2 + x - 2 = 0, since
12 + 1 - 2 = 0 .
rotation: A transformation in which a 2-D figure is
turned about a centre of rotation
run: The horizontal distance between two points
second differences: Values that are calculated by
subtracting consecutive first differences in a table of values
similar triangles: Triangles in which corresponding sides
are proportional and corresponding angles are equal
sine: The ratio of the length of the opposite side to the
length of the hypotenuse for either acute angle in a right
triangle; the abbreviation for sine is sin.
a
b
c
=
=
sine law: In any acute triangle,
sin A
sin B
sin C
C
b
a
A
c
B
slope: A measure, often represented by m, of the
steepness of a line; or the ratio that compares the vertical
and horizontal distances (called the rise and run) between
¢y
rise
=
two points: m =
run
¢x
solution to an equation: The value of a variable that
makes an equation true; for example, in the equation
5n + 4 = 39, the value of n is 7 because 5172 + 4 = 39.
solution to a system of linear equations: The values of
the variables in the system that satisfy all the equations;
for example, (7, 3) is the solution to x + y = 10 and
4x - 2y = 22.
solve for a variable in terms of other variables: The
process of using inverse operations to express one
variable in terms of the other variable(s)
square: A rectangle that has four equal sides
substitution strategy: A method in which a variable in
one expression is replaced with an equivalent expression
from another expression, when the value of the variable is
the same in both
supplementary angles: Two angles whose sum is 180°
S
scale factor: The value of the ratio of corresponding side
lengths in a pair of similar figures
system of linear equations: A set of two or more linear
equations with two or more variables; for example,
x + y = 10 and 4x - 2y = 22
scalene triangle: A triangle that has three sides of
different lengths
T
scatter plot: A graph that attempts to show a relationship
between two variables by means of points plotted on a
coordinate grid. It is also called a scatter diagram.
534
Glossary
table of values: An orderly arrangement of facts,
displayed in a table for easy reference; for example, a table
of values could be an arrangement of numerical values in
vertical and horizontal columns.
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tangent: The ratio of the length of the opposite side to
the length of the adjacent side for either acute angle in a
right triangle; the abbreviation for tangent is tan.
transformation: A change in a figure that results in
a different position, orientation, or size; for example,
a translation, a reflection, a rotation, a compression, a
stretch, and a dilatation are all transformations.
translation: A transformation in which a 2-D figure is
shifted left or right, or up or down; each point in the figure
is shifted the same distance and in the same direction.
transversal: A line that intersects or crosses two or more
lines
trapezoid: A quadrilateral that has one pair of parallel sides
trend: A relationship between two variables, with time as
the independent variable
trigonometry: The branch of mathematics that deals with
the properties of triangles and calculations based on these
properties
vertex (plural vertices): The point of intersection of a
parabola and its axis of symmetry
vertex form: A quadratic relation of the form
y = a(x - h)2 + k, where the vertex is (h, k)
vertical compression: A transformation that decreases all
the y-coordinates of a relation by the same factor
vertical stretch: A transformation that increases all the
y-coordinates of a relation by the same factor
volume: The amount of space that is occupied by an
object
X
x-intercept: The value at which a graph meets the
x-axis; the value of y is zero for all x-intercepts.
Y
y-intercept: The value at which a graph meets the
y-axis; the value of x is zero for all y-intercepts.
Z
V
zero principle: Two opposite integers that, when added,
give a sum of zero (for example, 1-12 + 1+ 12 = 0)
variable: A symbol used to represent an unspecified
number; for example, x and y are variables in the
expression x ⫹ 2y.
NEL
zeros of a relation: The values of x for which a relation
equals zero; the zeros of a relation correspond to the
x-intercepts of its graph.
Glossary
535
Glossary
trinomial: An algebraic expression that contains three
terms (e.g., 2x 2 ⫺ 6x y ⫹ 7)
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Answers
Chapter 1
c) slope: 0.5; y-intercept: 0
Getting Started, page 4
1.
2.
a) i
c) ii
b) iii, v, iv
d) vii
a)
y = 4x – 7
y
2
x
-4 -2
0
-2
2
e) vi
-4 -2
0
2
-2
4
8.
-8 (0, –7)
b) y = 2 – 3x
y
4
2 (0, 2)
x
4x – 5y = 10
y
2
(2.5, 0) x
-2
0
-2
2 4
(0, –2)
-2
-4
0
-2
2 ,0
—
3
(
4.
a)
4
-2
5.
2 4
-2
(0, –2)
-4
a) 2x + 22
b) 12x + 20
6.
7.
(6, 0)
0
6
2
x
-2
a)
3x + 4y - 6 = 0
3
3
y = - x +
4
2
b)
2x - y - 5 = 0
y = 2x - 5
c)
4x - 7y - 3 = 0
y =
d)
4x + 6y + 5 = 0
5
2
y = - x 3
6
-2
2
-4
-6
-8
536
Answers
0
-2
2
The relation in part c) is a direct relation because it is a straight line
that passes through the origin; the value of b in y = mx + b is 0. The
relations in parts a), b), and d) are partial variations because they are
straight lines that do not pass through the origin; the value of b in
y = mx + b is not 0.
a) 4.5 km
b) 18 km/h
a) linear; it is degree 1
b) linear; the first differences are constant
c) nonlinear; it is degree 2
d) nonlinear; the first differences are not constant
a) x = 7
c) x = 9
e) x = 10
b) x = - 4
d) x = 7
f) x = -2
a) - 4.8
b) about 2.01
a) Answers may vary, e.g.,
Factor out any
common factor.
or
Determine the
x-and y-intercepts.
e) 6x - 21
f ) 15x - 8
y = mx + b Form
-2
12.
13.
2
Ax + By + C = 0 Form
0
x
-2
x
0
c) -10x + 4
d) 5x + 1
y = 3x – 5
y
2
x
2
4
2
8
-2
a) slope: 3; y-intercept: - 5
9.
10.
11.
)
b) y = 5 – 2 x
y
(0, 5)
4
x – 3y = 6
y
2
-2
2
-4
-4
-6
a)
0
x
4
-4
3.
y = 2.6x – 1.2
y
4
x
-4 -2
y = –0.5x + 1.5
y
4
2 (0, 1.5)
b)
d) slope: 2.6; y-intercept: -1.2
y = 0.5x
y
2
4
3
x 7
7
2
b) slope: - ; y-intercept: 1
3
2
y = –—
3x +1
y
2
x
-2
0
-2
2
Determine the slope
and y-intercept.
Graph.
b) Answers may vary, e.g., I would determine the x- and y-intercepts
because I think it’s easier.
Lesson 1.1, page 12
1.
2.
a) on the graph because if x = 10, y = 0
b) not on the graph because if x = - 3, y = 6.5 not 7
c) on the graph because if x = 6, y = 2
d) on the graph because if x = 0, y = 5
e) on the graph because if x = 12, y = - 1
a)
Muffins
Doughnuts
Number Cost ($)
Number
Cost ($)
Total Cost ($)
0
0.00
60
15.00
15.00
4
3.00
48
12.00
15.00
8
6.00
36
9.00
15.00
12
9.00
24
6.00
15.00
16
12.00
12
3.00
15.00
20
15.00
0
0.00
15.00
NEL
1:11 PM
Page 537
b)
50
9.
40
30
20
10.
10
5.
Evening job (h)
10
0
7.
8.
x
10 20 30 40 50
Day job (h)
Rentals at Ben’s Bikes
y
120
80
40
x
0
40 80 120 160
Racing bikes
Answers may vary, e.g. let x represent euros, and let y represent Swiss
francs. Possible combinations are solutions to the equation
1.40x + 0.90y = 630; these can be shown in a graph or in a table
of values.
Euros (x)
Swiss Francs ( y)
0
700
90
560
700
180
420
600
270
280
360
140
450
0
800
20
500
400
300
100
10
0
Possible Combinations
of Francs and Euros
y
200
x
10 20 30 40
Day job (h)
No. Using the equations from question 5, Justin will earn $1580 per
week at the first job and $1520 per week at the second job.
a) Telephone Calls
Text Messages
Minutes Cost ($)
Number
Cost ($)
Total Cost ($)
250
25.00
0
0.00
25.00
190
19.00
100
6.00
25.00
130
13.00
200
12.00
25.00
70
7.00
300
18.00
25.00
10
1.00
400
24.00
25.00
0
0.00
416
24.96
24.96
0
12.
x
150
300 450
Euros
a) Let x represent the amount invested in the savings account,
and let y represent the amount invested in government bonds;
0.03x + 0.04y = 150
b)
Possible Incomes of
Bond and Account
y
5000
4000
Index
NEL
30
60 120 180 240
Number of minutes of calls
3000
2000
1000
0
Answers
20
11.
x
0
Glossary
6.
Answers may vary, e.g., I think the table of values is more useful
because it clearly lists some of Jacob’s options.
Answers may vary, e.g.,
a) (0, -1), (1, 4); (2, 5)
c) (0, 10), (2, - 40); (2, - 50)
b) (8, 0), (0, - 6); (10, 2)
d) (2, 10), (6, 0); (0, 0)
a) Let x represent the number of hours at the day job per week, and
let y represent the number of hours at the night job per week;
x + y = 40
b) Let x represent the number of hours at the day job per week, and
let y represent the number of hours at the night job per week;
15x + 11y = 540
c) Let x represent sales per week, and let y represent earnings per
week; y = 500 + 0.06x
d) Let x represent sales per week, and let y represent earnings per
week; y = 800 + 0.04x
e) Let x represent the number of nickels, and let y represent the
number of dimes; 0.05x + 0.10y = 5.25
Caroline’s Work Hours
Evening Hours vs.
a)
b)
y
Daytime Hours
50
y
50
40
40
30
Mountain bikes
4.
10 20 30
Muffins
Evening job (h)
3.
m
100
Appendix B
0
300
200
a) Let x represent sales, and let y represent earnings;
y = 1200 + 0.035x
b) Yes. Substituting x = 96 174 into the equation in part a) gives
y = 4566.09
a) 138
b) 115
c) Let x represent the number of racing bikes rented, and let y represent
the number of mountain bikes rented; 25x + 30y = 3450
Government bonds ($)
Doughnuts
60
Text Messages vs. Minutes
y
400
Appendix A
b) 20
c) 60
d) Answers may vary, e.g., 0.75x + 0.25y = 15; x represents number
of muffins bought, and y represents number of doughnuts bought.
e) Possible Combinations of
Muffins and Doughnuts
70 d
Swiss francs
5/13/09
Number of text messages
6706_PM10SB_Answers_pp536-546.qxd
x
2000 4000
Savings account ($)
Answers
537
6706_PM10SB_Answers_pp536-546.qxd
Page 538
Let x represent the registration fee, and let y represent the monthly
fee; x + 5y = 775
a) Answers may vary, e.g.,
Characteristics:
- degree 1
- constant first differences
- straight line graph
Methods of Representation:
- equation
- table of values
- graph
Linear Relation
Examples:
y 3x 14
y = 2x – 1
For y 2x,
x 1 2 3 4 5
y 2 4 6 8 10
6
6
4
4
2
2
2–4
y 3x2 3x 8
For y x2,
x 1 2 3 4 5
y 1 4 9 16 25
x
y
x
-6 -4 -2
0
-2
2
4
6
-6 -4 -2
0
-2
-4
-4
-6
-6
2
4
6
4.
b) Answers may vary, e.g.,
Method
equation
table of values
graph
Disadvantages
Situations to Use
• can be used to
determine any
values of x or y
• requires
arithmetic to
determine
answer
• when an exact
number (such
as one with
several decimal
places) is
needed
• shows possible
combinations
at a glance
• shows
information
visually
• does not list
all values
• may not show
a pattern
clearly
• may not be
accurate or
clear enough
to determine
exact values
5.
• when several
possible
combinations
are needed
quickly
6.
7.
8.
9.
120
80
40
x
40 80 120 160
Brazilian beans (kg)
1.
Answers may vary, e.g.,
a) about $125
b) about 200 km; about 350 km; about 150 km
2.
a) Let C represent the cost, and let d represent the distance;
3
d
C = 50 +
20
Answers
800
400
0
t
60 120 180 240300 360 390
Time after 10:00 a.m. (min)
b) 12:58 p.m.
a) A banquet for 160 people will cost $5700 at this hall.
b) about $7000
c) Answers may vary, e.g., let C represent the cost, and let n represent
the number of people; C = 32.5n + 500
d) 80, 120, and 138
e) Answers may vary, e.g., the data are discrete because the x-value
must be an integer.
Litres vs. Gallons
a)
40 L
24
16
0
g
2
4 6 8 10
U.S. gallons
b) Answers may vary, e.g., about 23 L
c) Answers may vary, e.g., about 3.5 gallons
4:59 p.m.
a) $15 500
b) Let E represent earnings, and let s represent sales;
E = 280 + 0.04s, 900 = 280 + 0.04(15 500)
a) 346 square feet
b) 7.5 square feet
a) Let V represent the value, and let t represent the time in years;
V = 4000 + (4000)(0.035)(t)
Value vs. Time
b) Answers may vary, e.g., about $4280
5000 V
c) 2.75 years or 2 years 9 months
4000
3000
2000
1000
0
Lesson 1.2, page 18
538
1200
8
Possible Combinations
of Beans
y
160
0
1600
32
• when showing
intercepts of
two different
relations
a) Answers may vary, e.g., you have $4.65 in dimes and quarters.
b) Answers may vary, e.g., you are paid a base salary of $900 and a
2.5% commission on sales.
Let x represent the amount of Brazilian beans, and let y represent the
amount of Ethiopian beans; x + y = 150, 12x + 17y = 14 * 150
Ethiopian beans (kg)
16.
Advantages
2000 V
Value ($)
15.
Juice Remaining
Non-examples:
y=x
y
3.
b) i) $125
ii) 200 km; about 333.33 km; about 166.67 km
c) Using an equation gave more accurate answers.
a) Let t represent the time in minutes after 10:00 a.m., and let
V represent the volume, in millilitres, remaining in the container;
V = 1890 - 5t
Volume remaining (mL)
14.
1:11 PM
Litres
13.
5/13/09
10.
11.
t
1 2 3
Time (years)
about 208 km
a) Write equations, make a table of values, or draw a graph.
b) If Cam has more than $12 000 in sales per week, he should stay at
his first job as he will earn more money there. Otherwise, he
should take the new job.
NEL
6706_PM10SB_Answers_pp536-546.qxd
13.
Page 539
Chantelle and Amit’s
Distance vs. Time
24 d
18 Amit
3.
x + y = 5, 3x + 4y = 12
y
a)
4
2
x
Chantelle
12
-2
6
t
0
0
2
-2
4
6
8 10
(8, –3)
-4
1
2
Time (h)
They will meet at 10:00 a.m.
Graphically, I would look along the horizontal line representing y = 5
until it intersected the line representing 2x - 3y = 6. Algebraically, I
would substitute y = 5 into the equation and solve for x.
once
Earnings vs. Sales
3000 E
b) (8, -3)
c)
Earnings ($)
2500
2000
4.
1500
1000
500
s
400
Cost of Buttons
300 Easyvans
200
Cost ($)
100 C
100
0
0
80
0
Cost ($)
of Buttons
All Seasons
x
300 600 900
Distance (km)
60
25
25
50
45
75
60
20
100
75
0
125
80
40
n
25 50 75 100 125
Number of buttons
c) I would recommend Easyvans if Alex was planning to drive more
than 667 km, because it would be cheaper. Otherwise, I would
recommend All Seasons because it would be cheaper.
a) x – y = 7, x + y = 3
c) 3x + y = 6
y
y = 2x – 4
2
y
x
2
0
x
2 4 6 8
-2
0
4
-2
-4
(5, –2)
-4 (2, 0)
-6
b) x + y = 8, 4x – 2y = 8
y
8
10
6
6
NEL
a) yes
a) i) (- 4, - 2)
b) i) (6, 5)
c) i) (5, - 4)
b) no
c) yes
d) no
ii) -4 = 2(- 2); 3( - 4) + ( - 2) = - 14
ii) 6 - 5 = 1; 6 + 3(5) = 21
ii) 2(5) - 5( -4) = 30; 5 + ( -4) = 1
-2
-4
2
4
6
8
4
2
(4, 2)
0
2
Index
1.
2.
x
0
2 x + y = 10
y =x –2
y
8
(4, 4)
4
2
Lesson 1.3, page 26
d)
Answers
b) Answers may vary, e.g., 100 buttons cost $75, and 101 buttons
cost $75.20. I would tell them to buy several extra now, while the
buttons are only $0.20 each. Then, if they needed more later, they
would not have to buy them at $1.00 each. This pricing structure
encourages people to buy more buttons since the more they buy,
the cheaper the price is per button.
c) Let C represent the cost in dollars, and let n represent the number
of buttons.
For n from 1 to 25, C = n.
For n from 26 to 50, C = 25 + 0.8(n - 25).
For n from 51 to 100, C = 45 + 0.6(n - 50).
For n from 101 or more, C = 75 + 0.2(n - 100).
5.
Glossary
16.
10 000 30 000 50 000
Sales ($)
about $38 333
a)
Number
Cost ($)
0
a) Let x represent the distance driven, and let y represent the total
cost in dollars; for Easyvans: y = 230 + 0.10x; for All Seasons:
y = 150 + 0.22x
b)
Cost vs. Distance
y
500
Appendix B
14.
15.
1:11 PM
Appendix A
Distance from start (km)
12.
5/13/09
x
-2
4
6
Answers
539
6706_PM10SB_Answers_pp536-546.qxd
e)
5/13/09
2
0
13.
a) Let x represent the number of student tickets, and let y represent the
number of non-student tickets; x + y = 679, 4x + 7y = 3370
b) 218
14.
a)
2
4
-2
Page 540
f ) 6x – 5y – 12 = 0
–2 x + 5y + 2 = 0
y
4
y = 3x – 5
y = –2 x + 5
y
6
-2
1:11 PM
x
(2, 1)
x
2
-2
4
0
2
4
-2
-4 (2.5, 0.6)
-4
-6
6.
Time driving at 50 km/h (h)
7.
8.
a) x + y = 80, 18x + 10y = 1000
b) 25 kg of pineapple mix, 55 kg of banana mix
80 kg of Brazilian beans, 120 kg of Ethiopian beans
a) Let x represent the time driving at 70 km/h, and let y represent the
time driving at 50 km/h; x + y = 6, 70x + 50y = 393
Time Driving
b)
y
10
9.
10.
11.
12.
4
2
x
x
2 4 6 8
b) 6 square units
a) Let C represent the cost, and let t represent the time of use;
for a regular bulb, C = 0.65 + 0.004t; for a fluorescent bulb,
C = 3.99 + 0.001t
b)
2 4 6 8 10
Time driving at 70 km/h (h)
16.
17.
12
Cotton price ($)
(1, 2)
0
4
c) 1.35 h at 50 km/h is 67.5 km
a) Let y represent earnings, and let x represent sales;
y = 1500 + 0.025x for Phoenix, y = 1250 + 0.055x
for Styles by Rebecca
b) about $8333
c) Joanna should take the job at Phoenix Fashions if she expects to
have monthly sales less than $8333 because it would pay more.
Otherwise, she should take the job at Styles by Rebecca because it
would pay more.
Answers may vary, e.g., at a fundraiser barbecue, hamburgers cost $3
and hotdogs cost $2. Martha is in charge of buying lunch for her class
of 25 students. Each student wants either a hot dog or a hamburger. If
Martha has collected $60 to spend, how many of each should she buy?
$11.01
a) Let x represent the price of denim fabric, and let y represent the
price of cotton fabric per metre; 3x + 5y = 22, 6x + 2y = 28
b)
Price of Fabrics
y
14
10
c) about 1113 h
d) The fluorescent bulb costs $22.94 less.
Answers may vary, e.g.,
• Read the problem, and determine what you need to find.
• Write two equations that describe the situation in the problem.
• Choose the best strategy to graph each equation.
• Graph both equations on the same set of axes.
• Label the graph.
• Determine the coordinates of the point of intersection.
• Verify the solution by substituting into both equations.
a) (3, -2)
b)
y
3x – y – 11 = 0
6
x + 2y + 1 = 0
4
9x + 4y – 19 = 0
2
x
-2
8
0
-2
6
-4
4
-6
(4, 2)
2
0
x
Answers
2
4
6
(3, –2)
-8
-10
2 4 6 8
Denim price ($)
c) $42
540
(4, 2)
2
15.
0
y =2
y = 4x – 2
y = –2 x + 10
(2, 6)
6
8
6
y
8
18.
19.
c) c = 2, d = 3
( -2, -5)
a) (1, 2), ( -2.5, 12.5)
b) about (2.618, 1.618)
NEL
6706_PM10SB_Answers_pp536-546.qxd
5/13/09
1:11 PM
Page 541
Mid-Chapter Review, page 32
a) Answers may vary, e.g.,
Possible Combinations of GIC
and Account Contributions
y
2500
Pears
Mass (kg)
Cost ($)
Mass (kg)
Cost ($)
Total Cost ($)
0.00
0.00
4.58
9.98
9.98
1.00
2.84
3.28
7.15
9.99
2.00
5.68
1.98
4.32
10.00
3.00
8.52
0.67
1.46
9.98
500
3.52
10.00
0.00
0.00
10.00
0
Pears (kg)
6.
7.
8.
2
0
2000
x
2 4
Apples (kg)
1500
1000
x
750 1500 2250
Account ($)
b) Answers may vary, e.g., about $625
c) $240
a) ( -1, 3)
b) (1.5, -1)
c) (1, 0)
d) (2, 0)
a) (2.2, 3)
b) (2.5, -3)
c) (4, 1.25)
d) (0.2, 0.5)
a) Let x represent student tickets, and let y represent non-student
tickets; x + y = 323, 2x + 3.50y = 790
b)
c) Let x represent the mass of apples, and let y represent the mass
of pears; 2.84x + 2.18y = 10.00
2.
Songs per
Month
Cost for
Site 1 ($)
Cost for
Site 2 ($)
0
12.95
8.99
15.20
13.74
17.45
18.49
15
19.70
23.24
20
21.95
27.99
Cost of Downloading Songs
y
Site 2
24
Cost ($)
30
18
Site 1
4.
x
4 8 12 16 20
Songs per month
a) If she has $1500 in sales, she will earn $360.
b) Answers may vary, e.g., about $425
c) Answers may vary, e.g., about $3700
d) Let y represent earnings, and let x represent sales; y = 300 + 0.04x;
for part b), y = 428; for part c), x = $3750
a) Answers may vary, e.g., let P represent the perimeter, and let x
represent the length; P = 2x + 2(x - 8)
b) 80 cm
2.
3.
4.
6.
7.
8.
NEL
Answers
541
Index
5.
a) y = 10x - 1 c) x = 20 - 2y
3
1
b) x = y d) y = 2x - 12
4
4
Let x represent the number of cars, and let y represent the number
of vans.
a) x + y = 53
b) 6x + 8y = 382
c) Answers may vary, e.g., x = 53 - y
d) Answers may vary, e.g., 6(53 - y) + 8y = 382; y = 32
e) Answers may vary, e.g., x + 32 = 53; 21 cars and 32 vans
a) ( -4, 3)
b) (2, 0)
3
a) b = 4 - 8a c) v = 3 - u e) y = 2x - 4
7
3
3
1
- s d) x = 6 + y
b) r =
f ) x = 15 - y
2
2
2
a) (7, 2)
c) (5, 4)
e) (4, -3)
8 4
b) (5, 1)
d) (4, 1)
f ) a- , b
3 3
registration fee: $120; monthly charge: $75
number of 500 g jars: 235; number of 250 g jars: 276
33°, 44°, 103°
Answers
1.
6
3.
10.
96 non-students
Yes. Answers may vary, e.g., the solution is (m + n, 3m + 2n).
Since m could be any integer and n could be any integer, the solution
could have a positive or negative x-value and a positive or negative
y-value. Therefore, the solution could be in any quadrant.
Answers may vary, e.g., Dan and Heidi are playing table tennis
against each other. After 17 points, Heidi is ahead by 7. How many
points has each player scored?
Lesson 1.4, page 38
12
0
9.
Glossary
5
10
Appendix B
b) Possible Combinations
of Pears and Apples
y
4
Appendix A
Apples
a) Let x represent the amount in the account, and let y represent the
amount in the GIC; 0.04x + 0.05y = 100
GIC ($)
1.
5.
6706_PM10SB_Answers_pp536-546.qxd
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
5/13/09
1:11 PM
Page 542
13
3
,y =
8
8
b) a = 2, b = 1
e) c = - 4, d = - 6
13
42
c) m = 13, n = - 35
f) x = - , y = 5
15
If Dan uses more than 12 cheques per month, then Save-A-Lot Trust
charges less. If he uses less than 12 cheques per month, then Maple
Leaf Savings charges less.
about 8.57 g of 80% silver, about 21.43 g of 66% silver
35 lawns
40 chairs, 5 tables
about 320.988 g of soy milk, about 345.679 g of vegetables
Nicole should accept the job at High Tech if she thinks she will make
less than $4000 per week in sales, because she will earn more.
Otherwise, she should accept the job at Best Computers because she
will earn more there.
a) In the second step, she incorrectly expanded - (4x - 10) as
-4x - 10. When solving for y, she calculated the value of 4( -7),
but did not subtract 10.
b) x = 3, y = 2; I substituted y = 4x - 10 into 2x - y = 4
and solved for x: 2x - (4x - 10) = 4, 2x - 4x + 10 = 4,
-2x + 10 = 4, x = 3; then I substituted x = 3 into
y = 4x - 10 and solved for y: y = 4(3) - 10, y = 2.
15 nickels, 27 dimes, 7 quarters
Answers may vary, e.g., I think this strategy is called substitution
because it involves two substitutions: the first substitution to obtain
an equation in only one variable, and the second substitution to solve
for the other variable.
x + 4y = 8 (equation 1)
3x - 16y = 3 (equation 2)
x = 8 - 4y (Isolate x using equation 1.)
3(8 - 4y) - 16y = 3 (first substitution)
3
y = (Solve for y.)
4
3
x = 8 - 4 a b (second substitution)
4
x = 5 (Solve for x.)
$160 000 in stocks, $100 000 in bonds, $40 000 in a savings account
a) x = - 2, y = 3
d) x =
4.
5.
6.
2.
3.
a) 3x - 2y = - 3, - x - 4y = 7
b) 3x + 2y = 21, - x - 4y = - 17
c) 4x - y = 11, 2x + 3y = - 5
d) 3x = 4, 5x + 4y = 12
13
9
a) i) a - , - b
iii) (2, - 3)
7
7
4 4
ii) (5, 3)
iv) a , b
3 3
13
9
13
9
b) i) 3a - b - 2 a - b = - 3, - a - b - 4a - b = 7
7
7
7
7
ii) 3(5) + 2(3) = 21, - (5) - 4(3) = - 17
iii) 4(2) - ( - 3) = 11, 2(2) + 3( -3) = - 5
4
4
4
iv) 3a b = 4, 5 a b + 4 a b = 12
3
3
3
a) (4, - 1)
b) 3x + y = 11, - x - 5y = 1
c) 3(4) + (- 1) = 11, - (4) - 5( -1) = 1
542
Answers
5
5
c) 10x + 15y = 25; a , 0 b and a 0, b
2
3
a) ( -1, 5)
b) 5x - y = - 10, 3x + 3y = 12
c) 5( -1) - (5) = - 10, 3( -1) + 3(5) = 12
a) 3x + 6y = 6, - 4x - 2y = 10
b) -x + 4y = 16, 7x + 8y = - 4
y
–x + 4y = 16
c)
6
3x + 6y = 6
4
7x + 8y = –4
–4x – 2y = 10
2
x
-6 -4 -2
0
-2
2
-4
-6
7.
8.
Lesson 1.5, page 46
1.
Answers may vary, e.g.,
a) 3x - 6y = 18
b) (6, 0) and (0, -3)
9.
10.
11.
12.
13.
14.
a) (4, 2)
b) 6x - 15y = - 6, - 5x + 15y = 10
c) x = 4, 11x - 30y = - 16
d) 6(4) - 15(2) = - 6, -5(4) + 15(2) = 10,
(4) = 4, 11(4) - 30(2) = - 16
a) Answers may vary, e.g., I don’t think it would affect the graph
because dividing by a non-zero constant is similar to multiplying
by its reciprocal, which is dividing by the non-zero constant.
Multiplying by a non-zero constant results in a proportional
increase for each term if the constant is greater than 1 and a
proportional decrease for each term if the constant is less than 1.
b) 8x + 4y = 4
c) x – y = 2
3x – 3y = 6
2x + y = 1
y
y
2
x
x
0
2
0
-2
-2
2
-2
(1, –1)
(1, –1)
Dividing the equations had no effect on the graph.
d) 3x = 3, -x - 2y = 1
e) 3(1) = 3, -(1) - 2( -1) = 1; they are equivalent.
1
1
a) a , 2 b
b) 2 a b + 11(2) = 23 c) a = 2, b = -3
2
2
Answers may vary, e.g.,
a) 2x - 5y = 9, 4x - 3y = - 3; 6x - 8y = 6, x + y = - 6
b) 2( -3) - 5(- 3) = 9, 4( - 3) - 3( -3) = - 3;
6(- 3) - 8( -3) = 6, ( -3) + (- 3) = - 6
Answers may vary, e.g.,
a) because either the x term or the y term will be eliminated
b) 4x - 3y = 10
c) 4x - 3y = 10
-4x - 2y = 3
7x - 3y = 12
2
4x = 12, 6y = - 4; a 3, - b
3
a) x + y = 33, x - y = 57
c) 45 and -12
b) 2x = 90, 2y = - 24
a) Let x represent the number of chicken dinners, and let y represent
the number of fish dinners; x + y = 200, 20x + 18y = 3880
b) 20x + 20y = 4000, 2y = 120
c) 140 chicken and 60 fish
NEL
6706_PM10SB_Answers_pp536-546.qxd
15.
17.
18.
Page 543
a) Yes. 3( -2) - 2( - 4) = 2, - 10(- 2) + 3( -4) = 8,
-7( -2) + (- 4) = 10, 13(-2) - 5( -4) = - 6
b) Answers may vary, e.g., 6x - 4y = 4, -14x + 2y = 20
Answers may vary, e.g.,
a) Equivalent systems of linear equations are systems that have the
same solution.
b) You can add them, subtract them, or multiply either one by a
non-zero constant.
c) This can sometimes help you solve the original system, by
cancelling out one of the variables.
a) 9x = - 18, -9y = - 45; ( -2, 5)
b) 23x = 46, -23y = 138; (2, - 6)
a) no
b) There are an infinite number of solutions.
c) The graphs are the same.
a) no
b) There is no ordered pair that represents a solution.
c) They are parallel and do not intersect at a point.
15.
16.
17.
18.
19.
Lesson 1.6, page 54
20.
1.
2.
21.
7.
y = 4x – 22
1
1
—
y = –—
3 – 3x
y
-2
8.
10.
11.
14.
NEL
Answers may vary, e.g.,
a) y = 3 x – 2
y = 3 x – 10
y
2
x
-2
0
2
-2
c) y = 3 x – 2
3y = 9x – 6
y
2
x
-2
4
0
-2
2
-4
x
-6
6
a) Let x represent the distance walked by Lori, and let y represent the
distance walked by Nicholas; x + y = 72.7, x - y = 8.9
b) x = 40.8, y = 31.9
a) Let l represent the length, and let w represent the width;
2l + 2w = 54, l - w = 9
b) l = 18, w = 9
about 276 g of the 99% cocoa, about 224 g of the 70% cocoa
a) (0.5, 3); 4(0.5) + 7(3) = 23, 6(0.5) - 5(3) = - 12
32
32
22
22
b) (22, 32);
= - 2,
= 3
11
8
2
4
c) (6, 5); 0.5(6) - 0.3(5) = 1.5, 0.2(6) - 0.1(5) = 0.7
( -1)
23
4
=
d) (4, - 1); - 5( -1) = 7, 3(4) +
2
2
2
1
1
1
e) a1, b ; 5(1) - 12a b = 1, 13(1) + 9 a b = 16
3
3
3
0 - 3
18
18
f ) (18, 0);
+
= 1,
- (0 + 9) = 0
9
3
2
30 g of mandarin orange, 40 g of tomato
3 km
3
2
and
3
4
-8
-10
b)
y = 3x – 2
y = –2 x + 6
y
6
4
2
x
-2
2.
0
2
-2
4
Answers may vary, e.g.,
a) i) 3x + 4y = - 3
ii) 5x + y = 9
iii) 6x + 8y = 4
b) i) Subtracting equations gives 0 = 5, which has no solution.
3x + 4y = 2
3x + 4y = –3
y
2
x
-2
0
-2
2
Answers
543
Index
12.
13.
-2
2 4
(5, –2)
1.
Answers
9.
0
Lesson 1.7, page 59
Glossary
3.
4.
5.
6.
a) subtract
b) subtract
c) subtract
d) add
a) I would multiply the first equation by 3 and the second equation
by 4. Then I would subtract one from the other.
b) I would subtract one from the other.
welder’s rate: $30/h; apprentice’s rate: $17/h
a) 2 and 1
b) 8 and 7
c) 5 and 3
d) 1 and 2
a) 3 and 4
b) 5 and 3
c) 1 and 2
d) 1 and 3
a) (- 2, 4)
c) (3, 7)
e) ( -5, 12)
b) (6, -1)
d) (0.5, 1)
f ) ( -0.2, 2.8)
(5, -2); My graph verifies my solution.
$2500 at 3%, $4000 at 4%
a) 45
b) 195
A
a) A + 8 = B + 2, + 18 = B + 9
2
b) A = 6, B = 12
Answers may vary, e.g., eliminating a variable means creating an
equation with the same solution, which has one less variable than the
original system.
3x + 7y = 31 (equation 1)
5x - 8y = 91 (equation 2)
Multiply equation 1 by 5 and equation 2 by 3, and subtract.
15x + 35y = 155
15x - 24y = 273
59y = - 118
Multiply equation 1 by 8 and equation 2 by 7, and add.
24x + 56y = 248
35x - 56y = 637
59x = 885
-5 and - 7
7
5
x = ,y =
5
2
de - bf
ce - af
a) x =
,y =
b) ad Z bc
ad - bc
bc - ad
Appendix B
19.
1:11 PM
Appendix A
16.
5/13/09
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5/13/09
1:11 PM
Page 544
e) infinitely many; 2x + 3y = 10
ii) 3(2) + 4(- 1) = 2, 5(2) + (- 1) = 9
3x + 4y = 2
5x + y = 9
y
2
x
-2
0
-2
2
4
iii) Subtracting the second equation from 2 times the first equation
gives 0 = 0.
f ) 1; (0, -0.4)
3x + 4y = 2
6x + 8y = 4
y
2
x
-2
3.
0
-2
2
a) 1; (- 11, - 38)
5 2
g) 1; a , b
6 3
b) 0; no solution
h) 0; no solution
c) infinitely many; y = 5x - 1.5
4.
d) 0; no solution
Answers may vary, e.g.,
a) y = 3x + 2, y = 3x - 3; subtracting equation 2 from
equation 1 gives 0 = 5.
y = 3x + 2
y = 3x – 3
y
4
2
x
-2
0
-2
2
-4
b) y = 2x, 3y = 8x; substitution gives (0, 0).
3y = 8x
y = 2x
y
2
x
-2
544
Answers
0
-2
2
NEL
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Page 545
c) y = x - 3, 3y = 3x - 9; substitution gives 0 = 0.
5.
x
-2
0
2
-2
4
-4
6.
Chapter Review, page 62
1.
Pounds
t
2 4 6
Time (h)
Cost ($)
Amount
Cost ($)
Total Cost ($)
0.00
0.00
350.00
700.00
700.00
200.00
300.00
200.00
400.00
700.00
400.00
600.00
50.00
100.00
700.00
466.67
700.00
0.00
0.00
700.00
Pounds
7.
8.
9.
10.
11.
400
300
0
x
150
300 450
Euros
Let x represent the amount in the savings account, and let y represent
the amount in the bond; 0.025x + 0.035y = 140
Possible Combinations of Bonds
and Savings Account Balances
y
6000
12.
13.
14.
15.
16.
17.
Bond ($)
5000
-4 -2
4000
3000
2
4
6
-4
Index
2000
-6
1000
0
0
-2
Answers
100
d) It represents the point where both companies charge the same
amount.
5
a) a - , 4 b
b) (15, 10)
c) (2, 4)
d) (2, 1)
2
registration fee: $150; monthly fee: $55
a) Let l represent the length, and let w represent the width;
2l + 2w = 40, l = w + 2
b) l = 11, w = 9
c) The rectangle is 11 m long and 9 m wide.
a) A
b) Answers may vary, e.g., x - 4y = 5, 3x - 6y = 3
c) Answers may vary, e.g., 4x - 10y = 8, -3x + 3y = 3
Answers may vary, e.g.,
a) x - 4y = 14, -5x - 2y = - 4;
-6x - 9y = 15, - 3x + y = - 9
b) (2) - 4( -3) = 14, -5(2) - 2( -3) = - 4;
-6(2) - 9( - 3) = 15, - 3(2) + (- 3) = - 9;
-2(2) - 3(- 3) = 5, 3(2) - ( -3) = 9
a) (2, -3)
b) ( -3, -1)
c) ( -2, 3)
d) (3, 12)
about 110 g of 99% cocoa, about 90 g of 70% cocoa
$150 in total; desk: $51, chalkboard: $99
34 37
a , b
11 11
36 $5 bills, 40 $10 bills
Answers may vary, e.g.,
a)
y
y = 2x + 4
4
y = 2x – 8
2
x
Glossary
Amount
200
NEL
20
0
Strategy 2: Let x represent the number of euros, and let y represent the
number of pounds; 1.5x + 2y = 700
Strategy 3:
Possible Combinations
of Pounds and Euros
y
500
3.
40
Strategy 1:
Euros
2.
a) Let C represent the cost, and let t represent the time; C = 20 + 8t
b) Let C represent the cost, and let t represent the time; C = 12 + 10t
c)
Cost of Renting
Snowblowers
c
60
Appendix B
6.
Answers may vary, e.g., if the coefficients and constants in both
equations are multiplied by the same amount, then there is an infinite
number of solutions. If the coefficients and constants in both
equations are not multiplied by the same amount, then there is one
solution. If the coefficients in both equations are multiplied by the
same amount but the constants are not, then there is no solution.
No. The linear system has no solution therefore the planes will not
collide.
Cost ($)
5.
Make a table of values, write equations, draw a graph; if you are
planning to drive more than about 333 km, then Bestcars is cheaper.
a) x = 2y + 2
b) 2 x – y = 1
x+y = 2
y –x = 1
y
y
4
2 (2, 0)
x
(2, 3)
2
0
2
x
-2
0
-2
2
-2
Appendix A
y = x –3
3y = 3x – 9
y
2
4.
-8
x
1500 3000 4500
Savings account ($)
a) After 5 h, there was 25.5 L of fuel left.
b) Answers may vary, e.g., about 37 L
c) Answers may vary, e.g., about 3:45 p.m.
18.
b) blue line: y = 2x - 8, 3y = 6x - 24;
red line: y = 2x + 4, 4y = 8x + 16
c) The slopes are equal.
a = 2.4, b = 4
Answers
545
6706_PM10SB_Answers_pp536-546.qxd
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Page 546
Chapter Self-Test, page 64
Let x represent the number of 500 g cartons, and let y represent the
number of 750 g cartons; 0.5x + 0.75y = 887.5
750 g bags
1.
1200
Possible Combinations
of Raisin Bags
y
2.
3.
4.
5.
6.
a) - 2
b) -1
6.
a) (1, 7)
7.
a) 6
8.
a) about 36.2 cm2
9.
400
trapezoid
a) Let V represent the volume remaining, and let t represent the time
after 8:30 a.m.; V = 1500 - 4t
b) Answers may vary, e.g., about 10:30 a.m.
c) 10:35 a.m.
- 24 1
a) (-1.5, 2.5)
b) a
, b
c) (2, - 3)
7 7
about 13.33 g of 85% gold, about 6.67 g of 70% gold
Answers may vary, e.g., at the point (x, y), which represents a solution
to a linear system, both sides of an equation in the system must be
equal. Therefore, adding or subtracting these equations is the same as
adding or subtracting constants to both sides of an equation: the
solution will remain the same.
a) 4x + 2y = 1, 2x + 6y = - 17; x = 2, y = - 3.5
b)
4x + 2y = 1, 2x + 6y = –17, x = 2,
y = –3.5, 3x + 4y = –8, x – 2y = 9
y
2
x
0
-2
2
4
parallelogram
rhombus
Lesson 2.1, page 78
1.
2.
3.
Getting Started, page 68
3.
4.
a) viii
c) ii
b) vii
d) v
a) 13 m
b) about 192.3 mm
1
14
a) y = x +
3
3
b) y = - 4x - 6
c) y = - 5x + 17
3
35
a) c) x
2
3
3
3
b) d) y
56
8
546
Answers
a) (3, 5)
b) ( -0.5, 3.5)
a) (2, 3)
b) (0.5, 1)
y
a)
A(11, 29)
30
B(32, 19)
20
10
6 km
$1500 in a savings account, $2700 in bonds
Answers may vary, e.g., adding the first equation to 3 times the
second equation and then simplifying gives 15 = 24, which is
not true.
e) iv
f ) vi
-10
f ) - 1.4375
c) (4, -2)
x
10 20 30 40 50
b) (32, 19)
a) (2, 5)
c) (2, -2)
e) ( -1, - 1)
b) (0.5, 3.5)
d) ( - 0.5, 0.5)
f ) (0.25, 0.75)
5. (0.75, - 1)
6. (5, 3); from P to M, run = 4 and rise = 2; the run and rise will be
the same from M to Q, so Q has coordinates (1 + 4, 1 + 2)
1
7. a), b) y = - x - 1
2
y
6
4.
4 C(0, 4)
g) i
h) iii
8.
2
0
E(1, 1) x
-2
2 4
A(2, –2)
-4
B(–4, –4)
-6
F(–1, –3)
-6 -4 -2
23
20
C(53, 9)
0
D(–2, 0)
e)
rectangle
square
6 8 10
(2, –3.5)
Chapter 2
2.
c) 0.7
quadrilateral
600 1200 1800
500 g bags
-4
1.
e) 3 or -3
f ) 8 or -8
x
-10 -8 -6 -4 -2
7.
8.
9.
c) 8
d) 6 or -6
3
b) a 1, b
2
1
b)
4
b) about 57.0 cm2, about 41.7 cm
800
0
5.
Answers may vary, e.g., ( -4, 4) and (2, 2) based on the assumption
that the centre is at O, or (5, 1) and ( - 1, 3) based on the assumption
that the centre is at R
NEL
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Page 546
Chapter Self-Test, page 64
Let x represent the number of 500 g cartons, and let y represent the
number of 750 g cartons; 0.5x + 0.75y = 887.5
750 g bags
1.
1200
Possible Combinations
of Raisin Bags
y
2.
3.
4.
5.
6.
a) - 2
b) -1
6.
a) (1, 7)
7.
a) 6
8.
a) about 36.2 cm2
9.
400
trapezoid
a) Let V represent the volume remaining, and let t represent the time
after 8:30 a.m.; V = 1500 - 4t
b) Answers may vary, e.g., about 10:30 a.m.
c) 10:35 a.m.
- 24 1
a) (-1.5, 2.5)
b) a
, b
c) (2, - 3)
7 7
about 13.33 g of 85% gold, about 6.67 g of 70% gold
Answers may vary, e.g., at the point (x, y), which represents a solution
to a linear system, both sides of an equation in the system must be
equal. Therefore, adding or subtracting these equations is the same as
adding or subtracting constants to both sides of an equation: the
solution will remain the same.
a) 4x + 2y = 1, 2x + 6y = - 17; x = 2, y = - 3.5
b)
4x + 2y = 1, 2x + 6y = –17, x = 2,
y = –3.5, 3x + 4y = –8, x – 2y = 9
y
2
x
0
-2
2
4
parallelogram
rhombus
Lesson 2.1, page 78
1.
2.
3.
Getting Started, page 68
3.
4.
a) viii
c) ii
b) vii
d) v
a) 13 m
b) about 192.3 mm
1
14
a) y = x +
3
3
b) y = - 4x - 6
c) y = - 5x + 17
3
35
a) c) x
2
3
3
3
b) d) y
56
8
546
Answers
a) (3, 5)
b) ( -0.5, 3.5)
a) (2, 3)
b) (0.5, 1)
y
a)
A(11, 29)
30
B(32, 19)
20
10
6 km
$1500 in a savings account, $2700 in bonds
Answers may vary, e.g., adding the first equation to 3 times the
second equation and then simplifying gives 15 = 24, which is
not true.
e) iv
f ) vi
-10
f ) - 1.4375
c) (4, -2)
x
10 20 30 40 50
b) (32, 19)
a) (2, 5)
c) (2, -2)
e) ( -1, - 1)
b) (0.5, 3.5)
d) ( - 0.5, 0.5)
f ) (0.25, 0.75)
5. (0.75, - 1)
6. (5, 3); from P to M, run = 4 and rise = 2; the run and rise will be
the same from M to Q, so Q has coordinates (1 + 4, 1 + 2)
1
7. a), b) y = - x - 1
2
y
6
4.
4 C(0, 4)
g) i
h) iii
8.
2
0
E(1, 1) x
-2
2 4
A(2, –2)
-4
B(–4, –4)
-6
F(–1, –3)
-6 -4 -2
23
20
C(53, 9)
0
D(–2, 0)
e)
rectangle
square
6 8 10
(2, –3.5)
Chapter 2
2.
c) 0.7
quadrilateral
600 1200 1800
500 g bags
-4
1.
e) 3 or -3
f ) 8 or -8
x
-10 -8 -6 -4 -2
7.
8.
9.
c) 8
d) 6 or -6
3
b) a 1, b
2
1
b)
4
b) about 57.0 cm2, about 41.7 cm
800
0
5.
Answers may vary, e.g., ( -4, 4) and (2, 2) based on the assumption
that the centre is at O, or (5, 1) and ( - 1, 3) based on the assumption
that the centre is at R
NEL
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1:12 PM
Page 547
9.
10.
12.
14.
15.
16.
20.
P(–4, 4)
-6 -4 -2
b) i)
12
T(–6, 12)
8
M(–3, 0)
-12 -8 -4
0
-2
2
4
x
6
y
8
ii) about 8.5 units
S(10, 2)
2
-2
x
0
4
-2 R(2, –1)
6
8 10 12
-4
-6
-8
c) i)
y
12
ii) 14 units
U(3.5, 11)
10
8
6
4
2
-2
x
0
2
-2
d) i)
4
6
8 10
T(3.5, –3)
-4
10
y
ii) 6 units
8
X(–1, 6)
Y(5, 6)
4
2
-4 -2
S(6, 6)
x
3.
0
4 8
N(3, –3)
4.
0
-2
x
2
4
6
8
a) about 403 km
b) I assumed that the helicopter flew in a straight line.
a) 3 units
c) about 6.1 units
e) about 8.2 units
b) about 6.3 units
d) about 6.1 units
f ) 8 units
U(0, –12)
Index
-12
Q(3, 1)
4
y
R(0, 9)
-8
4
6
4
-4
ii) about 7.6 units
6
2
equation of median from K: y = 8x - 11;
1
equation of median from L: y = - x ;
4
7
11
equation of median from M: y = x 5
5
5
3
7
15
a) y = x c) y = - x +
2
2
4
4
27
7
3
4
b) y = x d) y = x +
5
5
2
4
5
19
y = x 3
3
(4, 3)
a) y = 3x - 3
b) All the perpendicular bisectors have the equation y = 3x - 3.
c) They are the same. All the perpendicular bisectors are the same as
the line of reflection.
( - 2, - 2)
Answers may vary, e.g.,
x 1 + x 2 y1 + y2
i) Using the midpoint formula: (x, y) = a
,
b
2
2
ii) Using rise and run: Starting at one point, determine the rise and
the run. Then add half the run to the x-coordinate of this point
and add half the rise to the y-coordinate of this point.
These strategies are similar because they give the same answer. They
are different because you are using the mean value of x and y in part
i), but you are using the difference between x and y in part ii).
(4, 6); I added a third of the run from A to B to the x-coordinate of A,
and I added a third of the rise from A to B to the y-coordinate of A.
a) (0, 2)
b) (0, 2) for both medians
c)
8
c) about 6.4 units
d) 10 units
y
Answers
19.
2.
a) 5 units
b) 6 units
a) i)
Glossary
17.
18.
1.
Appendix B
13.
Lesson 2.2, page 86
Appendix A
yes
Answers may vary, e.g., in a rectangle, the diagonals bisect each other.
This means that the midpoint of one diagonal is also the midpoint of
the other diagonal. Mayda can determine the midpoint of the
diagonal for which she knows the coordinates of both endpoints.
Then she can use the coordinates of the midpoint and the coordinates
of the known endpoint of the other diagonal to determine the missing
coordinates.
11. a) M PQ = (2, 1), M QR = (1, -4), M RP = (6, 2)
b) slope of M PQ M QR: 5; slope of M QRM RP: 1.2; slope of
M RPM PQ: 0.25
c) slope of PQ: 1.2; slope of QR: 0.25; slope of RP: 5
d) Each midsegment is parallel to the side that is opposite it.
They intersect at (0, 2), which is two-thirds of the distance from
each point to the midpoint of the opposite side.
d) Answers may vary, e.g., yes, because I tried it on several different
triangles
NEL
Answers
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5.
a) i)
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y
10
1:12 PM
Page 548
8
A(2, 6)
2
B(5, 2)
0
b) i)
4
2
-2
6
6.
x
8
y
10
ii) about 6.3 units
8
7.
8.
6
C(–3, 4)
4
2
-6 -4 -2
9.
10.
11.
D(3, 2)
x
0
4
2
-2
c) i)
6
y
10
E(–6, 8)
ii) 17 units
8
12.
6
4
13.
2
14.
15.
16.
x
0
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
F(–6, –9)
ii) about 10.0 units
x
0
4
2
-2
6
a) The line segment in part f ) is horizontal because the
y-coordinates are the same. The line segment for part c) is
vertical because the x-coordinates are the same.
b) For a horizontal line, the length is the positive difference
between the x-coordinates. For a vertical line, the length is the
positive difference between the y-coordinates.
about 8.9 units
D; B is about 11.1 units from A, C is about 5.1 units from A, and
D is about 4.7 units from A.
the fire hall in Bently
Beast
Answers may vary, e.g., they are the same because they both use
the same information. They are different because the midpoint
formula uses the mean values of the x-coordinates and
y-coordinates, while the length formula uses the difference between
the x-coordinates and the difference between the y-coordinates.
a) about 4.1 units
c) about 7.2 units
b) about 5.7 units
d) about 1.7 units
8 6
1 27
Highway 2: a , b ; Highway 10: a , b
5 5
5 5
about 96.6 m
about 285.9 m
Answers may vary, e.g., I would use the distance formula,
d = 2(x 2 - x 1)2 + (y2 - y1)2, to calculate the distance
between each pair. For example, to determine whether A(3, 4) or
B( - 5, 2) is closer to C(1, 1),
Distance AC = 2(1 - 3)2 + (1 - 4)2 = 113 ⬟ 3.6
Distance BC = 2[1 - (- 5)]2 + (1 - 2)2 = 137 ⬟ 6.1
a) A¿ = (- 3, 0), B¿ = (0, 6), C ¿ = (4, 2)
b) No. Answers may vary, e.g., because the length of AB is not
equal to the length of DE
1.
-6
G(0, –7)
-8
4
y
ii) about 8.1 units
2
-4 -2
4 8 12
L(6, –2)
Lesson 2.3, page 91
8
-4
e) i)
17.
H(1, 3)
2
-4 -2
-10
y
4
x
0
-12 -8 -4
K(–10, –2) -4
4
d) i)
ii) 16 units
4
6
-4 -2
y
f ) i)
ii) 5 units
0
-2
I(–3, –3) -4
x
2
4
6
8
J(5, –4)
a) (7, 0), ( -7, 0)
c) 7
b) (0, 7), (0, -7)
d) x 2 + y 2 = 49
2. a) x 2 + y 2 = 9
b) x 2 + y 2 = 2500
49
c) x 2 + y 2 =
9
d) x 2 + y 2 = 160 000
e) x 2 + y 2 = 0.0625
3. a) i) 6 units
ii) (6, 0), ( - 6, 0), (0, 6), (0, - 6)
y
iii)
6
-6
4
-8
2
-6 -4 -2
0
-2
x
2
4
6
-4
-6
548
Answers
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6706_PM10SB_Answers_pp547-555.qxd
b) i) 7 units
iii)
1:12 PM
Page 549
ii) (7, 0), ( - 7, 0), (0, 7), (0, - 7)
y
8
iii)
6
y
iv)
20
0
4
-4
10
2
x
8
-6 -4 -2
-8
0
-2
y
x
2
4
6
-20 -10 0
-10
x
10 20
-20
-4
-6
c) i) 0.2 units ii) (0.2, 0), ( - 0.2, 0), (0, 0.2), (0, - 0.2)
y
iii)
0.4
8.
9.
x
0
-0.4
-0.4
d) i) 13 units
iii)
ii) (13, 0), ( - 13, 0), (0, 13), (0, - 13)
y
15
10
5
-15 -10 -5
0
-5
16.
17.
x
5 10 15
-10
4.
5.
a) x 2 + y 2 = 121
b) x 2 + y 2 = 81
a) 17 units
b) x 2 + y 2 = 289
c) x 2 + y 2 = 16
d) x 2 + y 2 = 36
y
c)
20
10
x
-20 -10 0
-10
10 20
18.
-20
4.
2
4
6
5.
2
x
-6 -4 -2
0
-2
-4
-4
-6
-6
x
2
4
6
6.
7.
8.
a) (4, 4)
(6, 3)
y = - 2x + 3
b) (1.5, 4)
c) (0, 4)
d) (3, 2)
7
39
x 2
4
a) M PQ = (3, 3), M QR = ( -5, 0), M RP = (4, 1)
7
1
b) y = - x +
10
5
c) y = - 9x + 30
a) about 5.4 units
c) 7 units
b) about 12.1 units
d) about 3.2 units
about 67.2 m
about 6.3 units
all points on the equation y =
Answers
Index
NEL
0
1.
2.
3.
4
2
-2
Mid-Chapter Review, page 95
iii) (3, 0), (3, 0)
iv) (8, 15), (8, 15)
y
ii)
6
4
-6 -4 -2
19.
c) yes, 82 + ( -1)2 = 65
d) no, (- 3)2 + ( -6)2 Z 65
iii) 3 units
iv) 17 units
iii) x 2 + y 2 = 9
iv) x 2 + y 2 = 289
Answers
a) yes, ( - 4)2 + 72 = 65
b) no, 52 + ( - 6)2 Z 65
7. a) i) 5 units
ii) 5 units
b) i) x 2 + y 2 = 25
ii) x 2 + y 2 = 25
c) Answers may vary, e.g.,
i) (3, 4), (3, 4)
ii) (0, 5), (5, 0)
y
d) i)
6
6.
Glossary
-15
Appendix B
10.
11.
12.
13.
14.
15.
0.4
a) x 2 + y 2 = 20 736
c) x 2 + y 2 = 1190.25
b) x 2 + y 2 = 361 000 000
d) x 2 + y 2 = 1.44
x 2 + y 2 = 9, x 2 + y 2 = 20.25, x 2 + y 2 = 56.25, x 2 + y 2 = 81,
x 2 + y 2 = 144
about 1257 km
a) x 2 + y 2 = 169
b) (-5, 12)
x 2 + y 2 = 289
about 37s
a = 10.0 or - 10.0, b ⬟ 6.6 or -6.6
Answers may vary, e.g., I would calculate the distance from (0, 0) to
(12 504, 16 050) and compare it with the square root of 45 000 000,
which is the radius of the first satellite’s orbit.
about 11.3 units by about 11.3 units
Answers may vary, e.g.,
Reason 1: The distance from the origin (0, 0) to the point (x, y) is
2(x - 0)2 + (y - 0)2, which is equal to 2x 2 + y 2. If, however,
(x, y) is a point on a circle with centre (0, 0) and radius r, then the
distance from (0, 0) to (x, y) is r. So 2x 2 + y 2 = r and, if you
square both sides, x 2 + y 2 = r 2.
Reason 2: The equation x 2 + y 2 = r 2 follows directly from the
Pythagorean theorem applied to the right triangle with vertices at
(0, 0), (x, 0), and (x, y).
Reason 3: Using the formula x 2 + y 2 = r 2, I can see that the x- and
y-intercepts are (r, 0), ( - r, 0), (0, r), (0, - r). This makes sense for a
circle, since all intercepts should be the same distance from the centre.
4
a) a circle centred at the origin, with a radius of
3
b) a circle centred at (2, 4), with a radius of 3
The most likely part of the load to hit the edge of the tunnel is the
corner. The distance from the middle of the road to the corner of the
load is 2(4)2 + (3.5)2 ⬟ 5.32, or about 5.32 m. Since this is larger
than the radius of the tunnel, the load will not fit through the tunnel.
Appendix A
4
4
-8 -4
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9.
10.
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length AB ⬟ 5.8 units, length BC ⬟ 8.2 units, length
CA ⬟ 8.6 units; so the sides are unequal
a) i) (0, 0)
ii) radius: 13 units; x-intercepts: (13, 0), (13, 0);
y-intercepts: (0, 13), (0, 13)
iii)
y
15
Page 550
Lesson 2.4, page 101
1.
2.
3.
10
4.
5
-15 -10 -5
x
0
-5
5 10 15
-10
5.
-15
6.
b) i) (0, 0)
ii) radius: 1.7 units;
x-intercepts: (1.7, 0), (1.7, 0);
y-intercepts: (0, 1.7), (0, 1.7)
y
iii)
2
1
-2 -1
1
-1
c) i) (0, 0)
ii) radius: about 9.9 units;
x-intercepts: about (9.9, 0), about (9.9, 0);
y-intercepts: about (0, 9.9), about (0, 9.9)
iii)
y
12
4
0
-4
y
4
x
4
x
0
2 4
-2 C(0, –2)
-4
b) scalene; GH = 126 ⬟ 5.10 units, HI = 120 ⬟ 4.47 units,
GI = 118 ⬟ 4.24 units; no equal sides
y
G(–1, 3)
4
2
I(2, 0) x
-4 -2
-2
H(–2, –2)
-4
2
4
8 12
-8
c) scalene; DE = 117 ⬟ 4.12 units, EF = 168 ⬟ 8.25 units,
DF = 5 units; no equal sides
-12
y
2
11.
A(3, 3)
B(–1, 2)
0
8
-12 -8 -4
1
1
= m VW
2
ABCD is a parallelogram. The lengths of the sides are needed to
determine if it is also a rhombus.
a) DE = FD = 165 ⬟ 8.06 units; EF = 1130 ⬟ 11.40 units
b) 132.5 ⬟ 5.72 units
7
1
c) m MD = =
m EF
9
PQRS is a parallelogram and a rhombus. The slopes of the sides or
the interior angles are needed to determine if it is also a square.
a) isosceles; AB = 117 ⬟ 4.12 units, BC = 117 ⬟ 4.12
units, CA = 134 ⬟ 5.83 units; two equal sides
2
-2
x
a) x 2 + y 2 = 25
b) x 2 + y 2 = 49
c) x 2 + y 2 = 73
d) x 2 + y 2 = 97
12. x 2 + y 2 = 900
13. a) on: 62 + ( - 3)2 = 45
b) outside: (- 1)2 + 72 7 45
c) inside: (- 3)2 + 52 6 45
d) outside: ( -7)2 + ( -2)2 7 45
14. a) 62 + ( - 7)2 = 85, 22 + 92 = 85
1
b) y = x
4
1
c) because (0, 0) is the centre of the circle and because 0 = (0),
4
(0, 0) also lies on the perpendicular bisector
-4 -2
0
4 6
D(2, –3)
E(–2, –4)
-6
F(6, –6)
-8
d) isosceles; JK = 158 ⬟ 7.62 units, KL = 6 units,
LJ = 158 ⬟ 7.62 units; two equal sides
6
y
J(2, 5)
4
2
0
Answers
2
-2
-2
L(–1, –2)
-4
550
1
4
The slopes are negative reciprocals: m TU = -
-4 -2
x
0
The slopes are equal: m PQ = m RS =
x
2
4 6
K(5, –2)
NEL
6706_PM10SB_Answers_pp547-555.qxd
7.
9.
11.
12.
13.
15.
20.
Answers may vary, e.g.,
a)
A
D
B
F
E
C
BC
AC
and ∠ ACB = ∠ FCE, ^ ABC is similar to
=
FC
EC
^ FEC. Each side in the larger triangle is twice the length of the
corresponding side in the smaller triangle.
Since
Answers
551
Index
NEL
Or, if the quadrilateral is both a rhombus and a rectangle, then it is a
square; e.g., J(3, 0), K(0, 4), L(- 4, 1), M(-1, -3).
a) rhombus; JK = KL = LM = JM = 217 ⬟ 4.12 units;
1
m JK = m LM = , m KL = m JM = 4; all sides are equal length, but
4
there are no right angles.
b) parallelogram; EF = GH = 5, FG = EH = 2153 ⬟
12.37 units; m FG and m EH are undefined (vertical), m FG = m EH
1
= , opposite sides are equal length and parallel, but there are no
4
right angles.
c) parallelogram; DE = FG = 250 ⬟ 7.07 units, EF = GH =
5
1
229 ⬟ 5.39 units; m DE = m FG = , m EF = m DG = ;
7
2
opposite sides are equal length and parallel, but there are no
right angles.
Answers
16.
1
d) rectangle; PQ = RS = 268 ⬟ 8.25 units, QR = PS =
= - 3, so PQ is perpendicular to PR; that is, PQ
m PR
1
217 ⬟ 4.12 units; m PQ = m RS = , m QR = m PS = - 4;
meets PR at a right angle.
4
1
4
opposite sides are equal length and parallel, and angles between
m MN = = - ; MN = NL = 241 ⬟ 6.40 units
m NL
5
sides are 90°.
a) Use the distance formula to determine the lengths of the three
17. square; all side lengths are 2106 ⬟ 10.30, or about 10.30 units, so
sides. If the lengths make the equation a 2 + b 2 = c 2 true, then
5 9 5 9
the side lengths are equal; slopes are , - , , - , so the slopes of
the triangle is a right triangle.
9 5 9 5
b) i) ^STU is a right triangle because
the sides are negative reciprocals.
ST 2 + TU 2 = 17 + 68 = 85 = US 2
18. a) S(8, 2)
ii) ^ XYZ is not a right triangle because XY 2 = 20, YZ 2 = 80,
b) Answers may vary, e.g., I determined the difference of the
and XZ 2 = 52
x- and y-coordinates between Q and P and then applied this
iii) ^ ABC is a right triangle because AB 2 + AC 2 = 13 + 13 =
difference to R.
26 = BC 2
c) yes, PR = QS = 2145 ⬟ 12.04, or about 12.04 units
a) WX = 229 ⬟ 5.39 units, XY = 241 ⬟ 6.40 units,
19. Answers may vary, e.g.,
YZ = 229 ⬟ 5.39 units, ZW = 241 ⬟ 6.40 units;
Determine
m WX = 0.4, m XY = - 1.25, m YZ = 0.4, m ZW = - 1.25
slopes and
b) parallelogram; because opposite sides are equal length and
lengths of sides.
parallel (since they have the same slope)
c) 290 - 250 ⬟ 2.42 units
2
4
Check for
Check for
one
Check for
, m = m UR = or RS = TU = 2104 ⬟
m RS = m TU =
none
equal sides.
parallel sides.
pair
equal sides.
10 ST
3
10.20, ST = UR = 5, so opposite sides are equal length and
parallel (because they have the same slope).
Non-parallel
None or only one
Two pairs of
Both pairs of
AB = BC = CD = DA = 5, so all sides are equal.
sides are
pair of adjacent
adjacent sides opposite sides none
equal.
sides are equal.
are equal.
are parallel.
a) EF = FG = GH = HE = 220 ⬟ 4.47, so all sides are equal;
1
m EF = m GH = - , m FG = m EH = 2, so adjacent sides meet
2
isosceles
Check for
at right angles.
trapezoid
irregular
kite
trapezoid
equal
sides.
1
b) m EG = - 3, m HF = ; the slopes of EG and HF are negative
3
Opposite sides
All sides
reciprocals, so EG and HF are perpendicular to each other.
are equal.
are equal.
7
3
1
m PQ = - , m QR = ; m PQ Z ; the slopes are not negative
m QR
9
2
Are adjacent
Are adjacent
reciprocals, so the sides are not perpendicular; that is, they do not
sides
sides
meet at right angles.
perpendicular?
perpendicular?
Answers may vary, e.g., I would use the distance formula to determine
the lengths of all the sides. If they are equal, then the quadrilateral is a
rhombus; e.g., A(3, 0), B(0, 2), C(- 3, 0), D(0, - 2). Or, I would use
yes
no
yes
no
the slope formula to determine the slopes of all the sides. If the slopes
of adjacent sides are negative reciprocals of each other, then the
rectangle
parallelogram
square
rhombus
quadrilateral is a rectangle, e.g., E(0, 0), F(2, 1), G(0, 5), H(-2, 4).
m PQ = -
Glossary
14.
Page 551
Appendix B
10.
1:12 PM
Appendix A
8.
5/13/09
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b)
5/13/09
1:12 PM
Page 552
B
13.
E
A
F
14.
H
D
G
C
^AHE is similar to ^ADB, so HE || DB.
^CGF is similar to ^CDB, so GF || DB.
^DGH is similar to ^DCA, so GH || CA.
^BFE is similar to ^BCA, so FE || CA .
Since HE || DB ||GF and GH || CA || FE, EFGH is a
parallelogram.
b) M BCM AD = 222.5 ⬟ 4.74, or about 4.74 units;
15.
Lesson 2.5, page 109
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
AC = BD = 265 ⬟ 8.06, so AC and BD are each about
8.06 units.
1
m JL = - , m KM = 2; the slopes of the diagonals are negative
2
reciprocals, so the diagonals are perpendicular.
Let S represent the midpoint of PR. Since S (1.5, 1.5), QS bisects
1
PR. Since m QS = = - 7, they are perpendicular.
m PR
M JL = M KM = ( -1, - 3)
AC = BD = 2520 ⬟ 22.80 units
Answers may vary, e.g., conjecture: Quadrilateral ABCD is a rectangle.
I calculated the length and slope of each blue line segment and found
that opposite sides are equal, and adjacent sides are perpendicular. My
conjecture was correct.
Answers may vary, e.g., conjecture: Quadrilateral JKLM is a kite. I
calculated the length of each blue line segment and found that
adjacent sides JK and KL are equal. Also adjacent sides JM and ML
are equal. My conjecture was correct.
Let G represent the midpoint of EF; G (0, 3). Since
1
= 5, DG is the median and the altitude.
m EF = m DG
Answers may vary, e.g., M PQ = ( - 1, 1), M QR = (3, 3.5), M RS =
(7, 1), M SP = (3, -1.5); M PQ M QR = M QR M RS = M RS M SP =
M SP M PQ = 222.25 ⬟ 4.72, or about 4.72 units; therefore,
M PQ M QR M RS M SP is a rhombus.
Answers may vary, e.g., M RS = ( -3, 2.5), M ST = ( -1.5, 1),
M TU = (- 4, -1.5), M UR = ( - 5.5, 0); diagonals M RS M TU =
M ST M UR = 217 ⬟ 4.12, or about 4.12 units; therefore, the
midpoints of the rhombus form a rectangle.
1
Answers may vary, e.g., m RT = = - 1; the slopes of the
m SU
diagonals are negative reciprocals, so the diagonals are perpendicular.
M RT = M SU = (3.5, 0.5), so the diagonals bisect each other.
a) (- 4)2 + 32 = 25, 32 + ( -4)2 = 25
b) m AB = - 1; therefore, the equation of the perpendicular bisector
of AB is y = x ; for y = x, when x = 0 and y = 0, left side equals
0 and right side equals 0 so the centre (0, 0) of the circle lies on the
perpendicular bisector.
a) M BC = ( -3, -0.5), M AD = (1.5, 1); slope of M BC M AD =
1
m AB = m DC = ; the slopes are the same, so the line segments
3
are parallel.
16.
210 + 240
BC + AD
=
= 1.5 210 = 222.5 ⬟ 4.74,
2
2
or about 4.74 units
1
Answers may vary, e.g., area of ^ABC = (7)(4) = 14, or
2
1
14 square units, area of ^M AB M BC M AC = (3.5)(2) =
2
1
3.5 = (14), or one-quarter of 14 square units
4
Answers may vary, e.g.,
Determine the midpoint of
the hypotenuse using the
midpoint formula.
Determine the distance from
the midpoint to each vertex
using the distance formula.
Compare the distances.
17.
Answers may vary, e.g., let the vertices of the square be
A(0, 0), B(2a, 0), C(2a, 2a), and D(0, 2a). The midpoints of
M AB M CD, M BC M AD, AC, and BD are all (a, a).
Lesson 2.6, page 113
1.
2.
(1, 4)
Answers may vary, e.g.,
A
C
3.
4.
B
E D
Let AD represent the median, and let AE represent the altitude.
Both ^ACD and ^ABD have the same base (since CD = DB)
and the same height, AE. Therefore, they have the same area.
a) AE * EB = CE * ED = 60
b) 1.25 m
a)
Answers may vary, e.g., M AB = ( - 4, - 10), M BC = ( -8, -2),
M CD = (0, 2), M DA = (4, -6); M AB M BC = M BC M CD =
M CD M DA = M DA M AB = 180 ⬟ 8.94 units, so the midsegments
form a rhombus. M AB M CD = M BC M DA = 1160 ⬟ 12.65, or
about 12.65 units, so the rhombus is a rectangle and, therefore, a square.
552
Answers
NEL
6706_PM10SB_Answers_pp547-555.qxd
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Page 553
b) Answers may vary, e.g.,
A1 ⴙ A2 (cm2)
square
36
64
100
100
semicircle
14.14
25.13
39.27
39.27
rectangle
18
32
50
50
equilateral triangle
15.59
27.71
43.30
43.30
right triangle
9
16
25
25
parallelogram
18
32
50
50
a) x 2 + y 2 = 85
b) 72 + 62 = 85
1
1
= , so the slopes are negative reciprocals;
c) m PR = m RQ
4
therefore, ∠ PRQ is a right angle.
8.
about 27 square units
16
a 2, b
3
16
a - , 0b
3
(2, 3)
(6, 4)
Answers may vary, e.g.,
i) Calculate DC, EC, and FC; check that they are equal.
ii) Construct the perpendicular bisectors for ^DEF; check that
their intersection is at C.
22
a , 8b
3
3:1
a) House A: about 37.2 m; house B: about 26.8 m; house C:
about 34.8 m; assuming that the connection charge is proportional
to distance, A has the highest charge.
b) about (18.1, 14.2)
Lesson 2.7, page 120
1.
2.
3.
4.
5.
6.
NEL
a)
1
2
b) –2
c) y = - 2x + 2
3
1
y = x 2
2
7 4
a ,- b
5 5
BC = 245 ⬟ 6.71, or about 6.71 units;
AD = 228.8 ⬟ 5.37, or about 5.37 units
18 square units
5
a) y = - x - 3
3
5
b) y = - x - 3
3
5
c) y = - x - 3
3
d) isosceles; the median and the altitude are the same.
9.
10.
11.
12.
13.
14.
15.
16.
Answers
553
Index
7.
Answers
c) no effect
d) The sum of the two smaller areas always equals the larger area.
Area on
Hypotenuse (cm2)
Glossary
A2: Area on
8 cm Side (cm2)
Appendix B
A1: Area on
6 cm Side (cm2)
Diagram
Appendix A
Similar
Figures
6706_PM10SB_Answers_pp547-555.qxd
17.
21.
Page 554
15.
B
A
20.
1:12 PM
right triangle; the lines forming two of the sides are perpendicular,
having slopes 1 and 1.
18.
19.
5/13/09
D
C
Use the chord intersection rule to find E such that AD * DC =
BD * DE; BD + DE = diameter.
Answers may vary, e.g., to determine the median, I would calculate
the midpoint of the opposite side and determine the slope of the
segment between the vertex and the midpoint, which gives m. Then I
would substitute one of these points into the equation of a line to
determine b. To determine the altitude, I would calculate the slope of
the opposite side and determine its negative reciprocal, which gives m.
Then I would substitute the vertex into the equation of a line to
determine b.
Let C represent the centroid;
3
5
4
M PQ = a , - 1 b, M QR = a , - 1 b, M RP = (0, 2), C = a , 0 b
2
2
3
QC
RC
PC
=
=
= 2
CM QR
CM RP
CM PQ
16.
17.
18.
19.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
(7.5, 28)
a) y = x + 2
1
b) yes, m median = = 1
m AC
7
21
y = x +
8
4
Q
about 89.5 units
about 21.8 units
10 units
a) x 2 + y 2 = 289
b) x-intercepts: (17, 0), (17, 0); y-intercepts: (0, 17), (0, 17);
points: e.g., (8, 15), (8, 15), (8, 15)
x 2 + y 2 = 841
x 2 + y 2 = 16
(-2)2 + k 2 = 20
k 2 = 16
k = 4 or 4
AB = BC = 217 ⬟ 4.12, or about 4.12 units; two sides are equal
in length, so the triangle is isosceles.
AB = 213 ⬟ 3.61, BC = 226 ⬟ 5.10, CA = 213 ⬟ 3.61, or
about 3.61 units; two sides are equal in length, so the triangle is
isosceles.
JK = KL = LM = MJ = 5 units; the sides are equal in length, so
the quadrilateral is a rhombus.
554
Answers
a) 62 + 72 = ( -9)2 + 22, so both points are the same distance
from (0, 0).
b) Let A represent point (6, 7). Let B represent point (–9, 2). Let C
represent the centre of the circle (0, 0). Let D represent the
3 9
intersection of the line and the chord. M AB = a - , b ;
2 2
1
m AB = ; m CD = - 3; therefore, the equation of the line through
3
Chapter Review, page 124
1.
2.
(10 - 5)2 + (10 + 2)2 = ( -7 - 5)2 + (3 + 2)2 = (0 - 5)2 +
(- 14 + 2)2
5
1
= ; the slopes of PQ and QR are negative
a) m PQ = m QR
2
reciprocals, so PQ and QR form a right angle.
5
b) M RP = a 2, b ; PM RP = QM RP = RM RP = 236.25 ⬟ 6.02,
2
or about 6.02 units
a) (3a)2 + a 2 = 10a 2, a 2 + ( - 3a)2 = 10a 2
b) Let C represent the centre; C = (0, 0), M RQ = (2a, -a); slope
1
RQ = 2, slope CM RQ = - ; the slopes are negative reciprocals.
2
Therefore, the line segments are perpendicular.
RS = TU = 220 ⬟ 4.47, ST = RU = 217 ⬟ 4.12, or about
4.12 units
1
1
m RS = 2, m ST = - , m TU = 2, m UR = 4
4
The opposite sides are equal and parallel, but the adjacent sides do
not meet at 90°. Therefore, the quadrilateral is a parallelogram.
M AB = ( -4, -4), M BC = (1, -5), M CD = (5, 1), M DA = (0, 2);
M AB M BC = M CD M DA = 226 ⬟ 5.10, or about 5.10 units;
M BC M CD = M DA M AB = 252 ⬟ 7.21, or about 7.21 units;
3
1
The slopes of the midsegments are and - . The opposite sides are
2
5
equal and parallel, but the adjacent sides do not meet at 90°.
Therefore, the quadrilateral is a parallelogram.
C is y = - 3x. Since M AB is on this line, D = M AB.
20.
21.
22.
23.
24.
25.
a) M JL = M KM = (5.5, -4.5)
b) Answers may vary, e.g., conjecture: Quadrilateral JKLM is a square.
c) Answers may vary, e.g., JK = KL = LM = MJ = 213 ⬟ 3.61,
or about 3.61 units; I calculated to determine that opposite sides
are equal and that adjacent sides are perpendicular. My conjecture
was correct.
(0.5, 2)
(3, 2)
about (19.9, 89.3)
Answers may vary, e.g., it will be a parallelogram because the slopes of
the lines form two pairs, which are not negative reciprocals;
4
4
m 1 = - 3, m 2 = , m 3 = - 3, m 4 =
5
5
453 194
,
b
a) a
17 17
b) about 6.79 m
Chapter Self-Test, page 126
1.
2.
a) about 40.3 m
11
b) a 4, b
2
a) x 2 + y 2 = 1296
b) 5 s
NEL
3.
5.
6.
7.
Page 555
7.
AB = BC = 5, or about 5 units, so two sides are equal;
3
1
m AB = = , so the slopes of two sides are negative
m BC
4
reciprocals.
a) Answers may vary, e.g., PR = QS = 221 125 ⬟ 145.3, or about
145.3 units
b) 390 units
rhombus
about (4.6, 51.4)
Answers may vary, e.g., to solve for the intercept, calculate the slope of
a line through the first two points, and then substitute one of these
points into the equation of a line. To determine the equation of the
new line, substitute the third point into the equation of a line with a
slope equal to the negative reciprocal of the first slope. To determine
the point of intersection, set these two equations equal to each other.
Calculate the distance between the point of intersection and the third
point.
right isosceles
8.
9.
Time (years)
0
1000
1
800
2
600
3
400
4
200
-6 -4 -2
a) v
c) i
b) ii
d) iv
a) about 58 beats per minute
b) about 38 years old
Answers may vary, e.g.,
Height of Baseball
a)
y
35
Height (m)
x
0
6
e) vi
f ) iii
(1.5, 0)
0
2
-2
4
2
x
6
-6 -4 -2
xy5 4
-2
-4
y
(0, 5)
6
0
2
4
6
-6 -4 -2
20
-4
15
-6
-6
c) x-intercept: ( - 2.5, 0);
y-intercept: (0, 5)
x
0
100
4
c) about 1.4 s and 3.6 s
c) -3x 3 + 6x 2 e) 14x 4 + 25x 3 - 44x 2 + 2x
d) - 7x 2 + 9x
f ) - 10x 4 + 40x 3 - 6x 2
b) 9
c) 23
d) -5
Cost of Airtime
Cost ($)
y
85
2
(–2.5, 0)
y
-2
6
(0, 5)
2
-2
x
4
6
4
6
y
4
y 2x 5
0
2
f ) x-intercept: none;
y-intercept: (0, - 5)
2
x
-6 -4 -2
(2, 0)
4
6
x
-6 -4 -2
0
-2
2
y 5
-4
-4
-6
-6 (0, –5)
25
75
35
45
300
55
400
65
500
75
600
85
a) false, e.g., y = x 2
b) false, e.g., y = 2x
55
45
35
25
0
x
y
1
1
2
4
x
3
9
200
400
600
Airtime (min)
4
16
First
Difference
3
5
7
x
y
1
2
2
4
3
6
4
8
First
Difference
Index
200
10.
65
Cost ($)
Airtime (min)
6
5
x2
2
2
2
C = 25 + 0.1t
NEL
Answers
b) about 32 m
a) 4x + 12
b) 2x 2 - 10x
a) 0
2 3 4
Time (s)
y
0
-2
1
6
3x 2y 12
2
(5, 0) x
-4
0
4
4
25
5
2
e) x-intercept: (2, 0);
y-intercept: none
2
-6 -4 -2
(4, 0) x
0
-6
b) x-intercept: (5, 0);
y-intercept: (0, 5)
6
y
(0, 6)
4
(0, –3)
-4
y 2x 3
-6
10
5.
6.
200
y
2
30
4.
400
Glossary
3.
800
600
4
Getting Started, page 130
2.
1000
1 2 3 4 5
5
0
Time (years)
V = 1000 - 200t
x-intercept: ( - 4, 0); y-intercept: (0, - 8)
a) x-intercept: (1.5, 0);
d) x-intercept: (4, 0);
y-intercept: (0, - 3)
y-intercept: (0, 6)
6
Chapter 3
1.
Value of Laptop vs. Time
y
Value ($)
Appendix B
8.
1:12 PM
Appendix A
4.
5/13/09
Value ($)
6706_PM10SB_Answers_pp547-555.qxd
Answers
555
3.
5.
6.
7.
Page 555
7.
AB = BC = 5, or about 5 units, so two sides are equal;
3
1
m AB = = , so the slopes of two sides are negative
m BC
4
reciprocals.
a) Answers may vary, e.g., PR = QS = 221 125 ⬟ 145.3, or about
145.3 units
b) 390 units
rhombus
about (4.6, 51.4)
Answers may vary, e.g., to solve for the intercept, calculate the slope of
a line through the first two points, and then substitute one of these
points into the equation of a line. To determine the equation of the
new line, substitute the third point into the equation of a line with a
slope equal to the negative reciprocal of the first slope. To determine
the point of intersection, set these two equations equal to each other.
Calculate the distance between the point of intersection and the third
point.
right isosceles
8.
9.
Time (years)
0
1000
1
800
2
600
3
400
4
200
-6 -4 -2
a) v
c) i
b) ii
d) iv
a) about 58 beats per minute
b) about 38 years old
Answers may vary, e.g.,
Height of Baseball
a)
y
35
Height (m)
x
0
6
e) vi
f ) iii
(1.5, 0)
0
2
-2
4
2
x
6
-6 -4 -2
xy5 4
-2
-4
y
(0, 5)
6
0
2
4
6
-6 -4 -2
20
-4
15
-6
-6
c) x-intercept: ( - 2.5, 0);
y-intercept: (0, 5)
x
0
100
4
c) about 1.4 s and 3.6 s
c) -3x 3 + 6x 2 e) 14x 4 + 25x 3 - 44x 2 + 2x
d) - 7x 2 + 9x
f ) - 10x 4 + 40x 3 - 6x 2
b) 9
c) 23
d) -5
Cost of Airtime
Cost ($)
y
85
2
(–2.5, 0)
y
-2
6
(0, 5)
2
-2
x
4
6
4
6
y
4
y 2x 5
0
2
f ) x-intercept: none;
y-intercept: (0, - 5)
2
x
-6 -4 -2
(2, 0)
4
6
x
-6 -4 -2
0
-2
2
y 5
-4
-4
-6
-6 (0, –5)
25
75
35
45
300
55
400
65
500
75
600
85
a) false, e.g., y = x 2
b) false, e.g., y = 2x
55
45
35
25
0
x
y
1
1
2
4
x
3
9
200
400
600
Airtime (min)
4
16
First
Difference
3
5
7
x
y
1
2
2
4
3
6
4
8
First
Difference
Index
200
10.
65
Cost ($)
Airtime (min)
6
5
x2
2
2
2
C = 25 + 0.1t
NEL
Answers
b) about 32 m
a) 4x + 12
b) 2x 2 - 10x
a) 0
2 3 4
Time (s)
y
0
-2
1
6
3x 2y 12
2
(5, 0) x
-4
0
4
4
25
5
2
e) x-intercept: (2, 0);
y-intercept: none
2
-6 -4 -2
(4, 0) x
0
-6
b) x-intercept: (5, 0);
y-intercept: (0, 5)
6
y
(0, 6)
4
(0, –3)
-4
y 2x 3
-6
10
5.
6.
200
y
2
30
4.
400
Glossary
3.
800
600
4
Getting Started, page 130
2.
1000
1 2 3 4 5
5
0
Time (years)
V = 1000 - 200t
x-intercept: ( - 4, 0); y-intercept: (0, - 8)
a) x-intercept: (1.5, 0);
d) x-intercept: (4, 0);
y-intercept: (0, - 3)
y-intercept: (0, 6)
6
Chapter 3
1.
Value of Laptop vs. Time
y
Value ($)
Appendix B
8.
1:12 PM
Appendix A
4.
5/13/09
Value ($)
6706_PM10SB_Answers_pp547-555.qxd
Answers
555
c) false, e.g., x = y 2 - 4
y
6
4
e) false, e.g.,
5.
6.
x y2 4
2
x
-6 -4 -2
0
2
-2
4
6
7.
-4
d) false, e.g., y = x 2 - 4
y
4
Lesson 3.2, page 145
1.
2
-6 -4 -2
x
0
2
-2
-4
4
6
y x2 4
-6
11.
2.
Answers may vary, e.g.,
3.
Definition:
- Any relation that is
not linear; any relation
that has terms with
power 2 or higher
Characteristics:
- Its graph is not a
straight line.
- First differences
are not constant.
Examples:
C ␲r2
x 0 1 2 3 4
y 1 3 9 27 81
Non-examples:
y 3x 4
x3
a 2b 3c d
4
x
-2
4.
5.
2
2
4
6
x
8 10
6.
Lesson 3.1, page 136
1.
2.
3.
4.
a) No, e.g., this is a straight line.
b) No, e.g., there are two openings.
c) Yes, e.g., the graph opens up in a U shape with a vertical line of
symmetry.
d) Yes, e.g., the graph opens down in a U shape with a vertical line of
symmetry.
e) No, e.g., the two arms of the graph are straight.
f ) No, e.g., the line segments are straight.
a) i) 1
ii) 2
iii) 2
iv) 3
b) ii) and iii), since they are degree 2
i) not a quadratic relation
iii) (0, 0)
ii) (0, 4)
iv) not a quadratic relation
a) first differences: 20, 20, 20; linear
b) first differences: 3, 5, 5; nonlinear; second differences: 2, 0
c) first differences: - 1, - 2, - 3; nonlinear; second differences:
-1, - 1; quadratic
d) first differences: - 2, 8, - 18; nonlinear; second differences:
10, -26
556
Answers
0
7.
2
-2
-4
4
0
y 0.08x(x 10)
2
y
6
a) y-intercept: 0; zeros: - 4, 0; vertex: ( -2, 4);
equation of the axis of symmetry: x = - 2
b) y-intercept: -3; zeros: -2, 2; vertex: (0, - 3);
equation of the axis of symmetry: x = 0
c) y-intercept: 0; zeros: 0, 4; vertex: (2, 4);
equation of the axis of symmetry: x = 2
a) maximum value: 4
c) maximum value: 4
b) minimum value: -3
y
8
6
Nonlinear
Relations
8
e) first differences: 1, 7, 19; nonlinear; second differences: 6, 12
f ) first differences: 1, 2, 4, 8; nonlinear; second differences: 1, 2, 4
a) downward
b) upward
c) downward
d) upward
a) upward, the coefficient of x 2 is positive
b) downward, the coefficient of x 2 is negative
c) downward, the coefficient of x 2 is negative
d) upward, the coefficient of x 2 is positive
The condition a Z 0 is needed. Without this condition, there would
be no x 2 term and the relation would be linear.
4
6
8 10 12
y 0.16x(x 10)
a) i) (3, 2)
iii) x = 3
ii) 0, 6
iv) negative, because the curve opens downward
b) i) (2, -3)
iii) x = 2
ii) 0, 4
iv) positive, because the curve opens upward
a) The vertex is a minimum because the curve opens upward.
b) The y-value is negative because there are two zeros and the vertex is
a minimum.
c) The x-value is 3; it must be halfway between the zeros.
a) i) x = - 4
iii) 0
v) - 8 (minimum)
ii) ( -4, - 8) iv) -8, 0
b) i) x = 2
iii) 0
v) 2 (maximum)
ii) (2, 2)
iv) 0, 4
c) i) x = 0
iii) 8
v) 8 (maximum)
ii) (0, 8)
iv) -6, 6
a)
y x2 2
x
y
y
-2
6
8
-1
3
6
0
2
4
1
3
2
2
6
0
i) x = 0
ii) (0, 2)
iii) 2
iv) none
-4 -2
(0, 2)
2
x
4
v) 2 (minimum)
NEL
6706_PM10SB_Answers_pp556-567.qxd
b)
x
-2
-4
0
-1
-6
1
-2
-8
2
-5
8
-1
3
0
0
1
-1
2
0
-1
-5
0
0
1
3
2
4
3
3
4
0
-4 -2
1
0
2
1
3
4
y
-3
0
-2
3
-1
4
3
1
0
i) x = - 3
ii) ( -3, -1)
iii) 8
iv) -4, -2
v) -1 (minimum)
i) x = 3
ii) (3, 4)
iii) -5
iv) 1, 5
v) 4 (maximum)
i) x = 2
ii) (2, - 8)
iii) 0
iv) 0, 4
v) - 8 (minimum)
-4
v) -1 (minimum)
c)
y x2 4x
y
(2, 4)
4
2
(0, 0)
-2
(4, 0) x
0
2
-2
4
6
-4
v) 4 (maximum)
d)
y x2 2x 1
y
6
4
2
(1, 0) x
-2
0
2
-2
4
v) 0 (minimum)
e)
y x2 2x 3
y
(–1, 4)
(–3, 0)
-4 -2
iii) 3
iv) -3, 1
2
0
-2
(1, 0) x
2
v) 4 (maximum)
Answers
557
Index
0
v) 4 (maximum)
2 4
-2 (1, –1)
iii) 1
iv) 1
x
iii) 4
iv) -2, 2
(2, 0) x
0
Answers
4
1
i) x = - 1
ii) ( -1, 4)
NEL
2
(0, 0)
y
0
i) x = 0
ii) (0, 4)
4
iii) 0
iv) 0, 4
-1
v) -1 (minimum)
Glossary
y
iii) 3
iv) 1, 3
b)
y x2 2x
y
6
iii) 0
iv) 0, 2
x
i) x = 2
ii) (2, -1)
Appendix B
-2
x
y x2 1
v) -1 (maximum)
iii) -1
iv) none
y
a)
Appendix A
-1
x
8.
2 4
-2 (0, –1)
-5
i) x = 1
ii) (1, 0)
f)
-4 -2
x
0
-2
i) x = 2
ii) (2, 4)
e)
Page 557
y
i) x = 1
ii) (1, - 1)
d)
1:14 PM
y
i) x = 0
ii) (0, - 1)
c)
5/13/09
f)
9.
10.
11.
12.
13.
14.
15.
16.
17.
ii)
i) x = 4
iii) 0
ii) (4, 8)
iv) 0, 8
a) x = 6
b) x = - 5.5 c) x = - 0.75
d) x = - 3
(2, 5)
a) Disagree, e.g., not all parabolas intersect the x-axis.
b) Agree, e.g., all parabolas intersect the y-axis once.
c) Disagree, e.g., all parabolas that open downward have negative
(constant) second differences.
a) 0, 4; 4 s
b) (2, 20)
c)
a) If a is negative, the quadratic relation has a maximum; if a is
positive, it has a minimum.
b) The y-value of the vertex is the maximum or minimum value
of the parabola.
a) i)
558
y
x
1
2
2
8
3
18
4
32
5
50
Answers
First
Difference
Second
Difference
6
10
14
18
4
4
4
y
1
2
2
4
3
8
4
16
x
y
1
1
2
8
3
27
4
64
x
y
iii)
v) 8 (maximum)
d) 20 m; 2 s
a) $60 000
b) 1000
c) Yes. There are two break-even points: 0 players and 2000 players.
a) 500 m
b) 10 s
c) 320 m
d) about 8.9 s
Profits increased (or losses decreased) if less than 900 000 game
players were sold, since the y-value of the line representing this
year’s profit is higher for x 6 9. If more than 900 000 game
players were sold, then losses increased.
x
2
4
4
8
2
2
32
3
162
4
512
Second
Difference
7
12
19
18
37
First
Difference
1
Second
Difference
2
First
Difference
iv)
18.
First
Difference
Second
Difference
30
100
130
220
350
b) Yes. The relation in part i) represents a parabola because it has
constant second differences.
c) Yes. The relation in part i) is quadratic because it has constant
second differences.
Answers may vary, e.g., I looked at several examples and compared the
x-coordinate of the vertex with the coefficient of x :
1
y = 5x 2 - 4.2x + 8; 0.42: - of the coefficient of x
10
1
y = 2x 2 - 3.2x + 8; 0.64: not - of the coefficient of x
10
1
y = 5x 2 - 3.2x + 6; 0.32: - of the coefficient of x
10
I made the hypothesis that the x-coordinate of the vertex will equal
1
- of the coefficient of x whenever the coefficient of x 2 equals 5. To
10
prove my hypothesis, I investigated the equation y = 5x 2 + bx,
b
where c = 0. Since the zeros of this equation are 0 and - ,
5
b
the x-coordinate of the vertex is - . Different values
10
of c will change the y-intercept, but not the line of symmetry
or the x-coordinate of the vertex.
Lesson 3.3, page 155
1.
a) i) 0, -3
ii) The zeros are the values of x where a factor equals 0.
iii) 0
iv) x = - 1.5
v) ( -1.5, 4.5)
vi) Yes. The highest power of x in the expanded form is 2.
NEL
6706_PM10SB_Answers_pp556-567.qxd
vii)
6
(–1.5, 4.5)
-6 -4 -2
1:14 PM
Page 559
y
5.
a)
y (x 3)(x 3)
y
4
4 y 2x(x 3)
2
(0, 0)
x
(–3, 0)
0
-6 -4 -2
-2
2
4
Appendix A
(–3, 0)
5/13/09
2
6
(3, 0) x
0
2
-2
-4
-4
-6
-6
4
6
-8
(0, –9)
-10
b)
y (x 2)(x 2)
y
10
8
6
4
(–1, 0)
-6 -4 -2
4 (0, 4)
2
-2
-4
2
(–2, 0)
4 6
y (x 3)(x 1)
-8 -6 -4 -2
(1, –4)
c)
-6
c) i) -2, 1
ii) The zeros are the values of x where a factor equals 0.
iii) -4
iv) x = - 0.5
v) ( -0.5, - 4.5)
vi) Yes. The highest power of x in the expanded form is 2.
y
vii)
6
y 2(x 1)(x 2)
4
0
-2
-4
2
2
x
(2, 0)
-4 -2
0
2
-2
4
6
8
y (x 2)(x 2)
y
6
(0,
4)
4
2
(–0.5, –4.5)
-6 -4 -2
2.
(2, 0)
0
2
-2
x
4
Answers
(–2, 0)
-6
6
-4
-6
e)
y 2(x 3)2
20
y
16 (0, 18)
12
8
4
x
(–3, 0)
-8 -6 -4 -2
0
Index
NEL
y (x 2)(x 2)
y
10
4 (0, 4)
6
a) ii
b) iv
c) i
d) vi
e) v
f ) iii
5
3.
9
4. a) y-intercept: - 9; zeros: - 3, 3; equation of the axis of symmetry:
x = 0; vertex: (0, - 9)
b) y-intercept: 4; zero: -2; equation of the axis of symmetry: x = - 2;
vertex: ( -2, 0)
c) y-intercept: 4; zero: 2; equation of the axis of symmetry: x = 2;
vertex: (2, 0)
d) y-intercept: 4; zeros: -2, 2; equation of the axis of symmetry:
x = 0; vertex: (0, 4)
e) y-intercept: 18; zero: -3; equation of the axis of symmetry:
x = - 3; vertex: ( - 3, 0)
f ) y-intercept: - 64; zero: 4; equation of the axis of symmetry: x = 4;
vertex: (4, 0)
-2
4
8
x
4
2
Glossary
-6 -4 -2
(1, 0)
x
0
6
d)
2
(–2, 0)
2
x
(3, 0)
0
Appendix B
b) i) -1, 3
ii) The zeros are the values of x where a factor equals 0.
iii) –3
iv) x = 1
v) (1, - 4)
vi) Yes. The highest power of x in the expanded form is 2.
y
vii)
6
2
Answers
559
6706_PM10SB_Answers_pp556-567.qxd
f)
y 4(x 4)2
y
10
(4, 0)
0
-2
-10
2
4
6
5/13/09
1:14 PM
Page 560
11.
x
8 10
-20
The equation y = a(x - r)(x - s) cannot be used if there are no
zeros. This occurs when a quadratic of the form y = ax 2 + bx + c
cannot be factored. Two examples are y = x 2 + x + 1 (blue) and
y = - 2x 2 - 2x - 4 (red).
y x2 x 1
y
10
-30
8
-40
6
-50
4
-60
2
(0, –64)
-70
x
-6 -4 -2
6.
7.
8.
3
1
1
a)
b)
c) 1.6
d) e) - 0.4
8
8
8
a) zeros: -40, 40; equation of the axis of symmetry: x = 0;
1
vertex: (0, 40); equation: y = - (x - 40)(x + 40)
40
b) zeros: 10, 30; equation of the axis of symmetry: x = 20;
1
vertex: (20, - 10); equation: y =
(x - 10)(x - 30)
10
c) zeros: -4, 1; equation of the axis of symmetry: x = - 1.5;
vertex: ( -1.5, - 2); equation: y = 0.32(x + 4)(x - 1)
d) zeros: -1, - 5; equation of the axis of symmetry: x = - 3;
vertex: (- 3, 3.5); equation: y = - 0.875(x + 1)(x + 5)
a)
y
12
6
(–3, 0)
-4 -2
0
-6
4
6
-8
-10
y 2x2 2x 4
12. a), b)
y 5(x 1)(x 5)
y
50
(2, 45)
40
30
20
10
(–1, 0)
-2
4
y 3(x 2)(x 3)
-18 (–0.5, –18.75)
13.
b) a = 2: vertex shifts up, curve is wider;
a = 1: vertex shifts up farther, curve is even wider;
a = 0: straight line along x-axis;
a = - 1: reflection of curve when a = 1 about x-axis;
a = - 2: reflection of curve when a = 2 about x-axis;
a = - 3: reflection of curve when a = 3 about x-axis
a)
y
8
2
-4
-12
9.
-2
-6
(2, 0) x
2
0
14.
15.
16.
0
(0, 25)
x
(5, 0)
2
4
6
c) - 1 s d) - 1, 5 e) (2, 45) f ) y = - 5(x + 1)(x - 5)
g) (5, 0) represents the point where the ball hits the ground; ( - 1, 0)
represents negative time and has no physical meaning.
1
a) y = (x + 250)(x - 50)
2500
b) The price should be decreased by $100.
$13
a) y = (5 + 0.1x)(700 - 10x)
b) $3600
c) 600
Answers will vary, e.g.,
Determine the two
zeros: r and s.
6
4
2
-2
10.
0
-2
y (x 2)(x 3)
(2, 0)
x
(3, 0)
2 4 6
(2.5, –0.25)
Determine the
y-intercept: c.
8
Calculate a c r s.
b) The second zero would move from 3 to match each value of s.
The farther the second zero was from 2, the lower the vertex
would shift.
a) y = 5(x + 3)(x - 5)
b) (1, -80)
Write the equation as:
y a(x r)(x s).
17.
18.
560
Answers
Expansion gives
a) iv
b) i
12 m by 24 m
c) ii
d) v
NEL
6706_PM10SB_Answers_pp556-567.qxd
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1:14 PM
Page 561
Mid-Chapter Review, page 160
2.
3.
a) Yes. The graph has a U shape.
b) Yes. The degree of the equation is 2.
c) No. The second differences are not a non-zero constant.
a) upward
b) downward
y x2 6x
y
10
(3, 9)
8
a) y-intercept: - 25; zeros: - 5, 5;
equation of the axis of symmetry: x = 0; vertex: (0, - 25)
y
10
-6 -4 -2
y (x 5)(x 5)
-25 (0, –25)
2
4
6
8
b) (3, 9)
c) 0
Appendix B
a) x = 3
x = 3
a)
6
-30
(6, 0) x
(0, 0)
4.
5.
-5
x
4
-20
2
0
2
-15
4
-2
(5, 0)
0
-10
6
-2
5
(–5, 0)
Appendix A
1.
7.
b) y-intercept: - 12; zeros: 2, 6;
equation of the axis of symmetry: x = 4; vertex: (4, 4)
y
(4, 4)
4
d) 0, 6
2
-2
0
-2
(2, 0)
2
4
(6, 0)
6
x
8 10
y (x 6)(x 2)
-4
-6
-8
y-intercept: 15;
zeros: - 3, - 5;
equation of the axis of symmetry: x = - 4;
vertex: ( - 4, - 1)
-12
(0, –12)
c) y-intercept: - 6; zeros: - 3, 1;
equation of the axis of symmetry: x = - 1; vertex: ( - 1, - 8)
y
2
(–3, 0)
(1, 0)
x
b)
-6 -4 -2
0
-2
-4
2
4
Glossary
-10
6
y 2(x 1)(x 3)
-6 (0, –6)
(–1, –8)
Answers
6.
-10
y-intercept: - 32;
zero: 4;
equation of the axis of symmetry: x = 4;
vertex: (4, 0)
a) 0.5 m
b) about 3 s
c) 1.5 s
d) 11.75 m
e) 0.5 m; the ball is travelling downward because this is after it has
reached its maximum height.
f ) about 0.9 s and 2.1 s
d) y-intercept: - 8; zero: - 4;
equation of the axis of symmetry: x = - 4; vertex: ( - 4, 0)
y
2
(–4, 0)
x
-10 -8 -6 -4 -2
0
-2
2
-4 y 0.5(x 4)2
-6
(0, –8)
-10
9.
10.
NEL
Index
8.
1
200
b
a) y = - (x - 30)(x + 10)
b) a10,
6
3
y = - 2(x + 2)(x + 6)
Answers may vary, e.g., y = (x + 5)(x + 5)
Answers
561
Lesson 3.4, page 165
1.
4.
a) x + 1, x + 5; x 2 + 6x + 5
b) x - 2, x - 2; x 2 - 4x + 4
Expression
2.
a)
5.
Area Diagram
x 2 + 7x + 6
x
1
x
x2
x
6
6x
6
(x + 1)(x + 6)
Expanded and
Simplified Form
6.
7.
8.
9.
b)
(x + 1)(x - 4)
x
1
x
x2
x
4
4x
4
x 2 - 3x - 4
10.
11.
12.
c)
d)
x
2
x
x2
2x
2
2x
4
x
3
x2
3x
(x - 2)(x + 2)
(x - 3)(x - 4)
x
x2 - 4
x 2 - 7x + 12
13.
14.
4
e)
f)
4x
12
x
2
x
x2
2x
4
4x
8
x
2
x
x2
2x
6
6x
12
(x + 2)(x + 4)
(x - 2)(x - 6)
15.
x 2 + 6x + 8
16.
17.
x 2 - 8x + 12
18.
19.
3.
a) m 2, 6
b) k 2, k
562
Answers
c) 4r, 12
d) 5x, 2x
e) 6n 2, 4n
f ) 15m, 6
a) x 2 + 7x + 10
d) x 2 + x - 2
b) x 2 + 3x + 2
e) x 2 - 6x + 8
c) x 2 - x - 6
f ) x 2 - 8x + 15
a) 5x 2 + 12x + 4
d) 3x 2 + x - 2
b) 4x 2 + 9x + 2
e) 4x 2 - 14x + 12
c) 7x 2 - 11x - 6
f ) 7x 2 - 26x + 15
a) x 2 - 9
d) 9x 2 - 9
b) x 2 - 36
e) 16x 2 - 36
c) 4x 2 - 1
f ) 49x 2 - 25
a) x 2 + 2x + 1
d) 25y 2 - 20y + 4
b) a 2 + 8a + 16
e) 36z 2 - 60z + 25
c) c 2 - 2c + 1
f ) 9d 2 - 30d + 25
a) 8m 2 + 4m - 12
c) 10x 2 - 14x - 12
b) 9m 2 + 12m + 4
d) 4x 2 + 6x + 2
a) 4x2 + 4x - 168
d) 2x 2 + 6x - 13
b) -4x 2 - 11x + 3
e) 15x 2 - 6x - 10
c) 6x 3 + 12x 2 + 6x
f ) 15x 2 + 60x + 20
a) 2x 2 + 5x y + 3y 2
d) 56x 2 + 9xy - 2y 2
b) 3x 2 + 7x y + 2y 2
e) 36x 2 - 25y 2
c) 15x 2 + 2x y - 8y 2
f ) 81x 2 - 126x y + 49y 2
a) y = x 2 - 2x - 8
c) y = 2x 2 - 8x
b) y = - x 2 - 6x - 8
d) y = - 0.5x 2 + 4x - 6
15
35
5
x +
a) - x 2 +
; downward
16
8
16
b) x2 + 6x + 5; upward
1
10
c) x 2 x + 3; upward
7
7
4
12
1
d) x 2 - x ; upward
7
7
7
7 2
42
112
e) x +
x +
; downward
25
25
25
Agree, e.g., there is a common factor of 2, which can be applied to
either of the other factors.
The highest degree of each of the two factors is 1. Therefore, their
product will have degree 2.
176
88 2
x +
x
Answers may vary, e.g., y = 1764
42
No, e.g., sometimes the like terms have a sum of zero and the result is
binomial; two examples are (x + 2)(x - 2) = x 2 - 4 and
(3x + y)(3x - y) = 9x 2 - y 2.
a) x 3 + 9x 2 + 27x + 27
b) 8x 3 - 24x 2 + 24x - 8
c) 64x 3 + 96x 2y + 48x y 2 + 8y 3
d) x 4 - 8x 2 + 16
e) x 4 - 45x 2 + 324
f ) 9x 4 + 36x 3 + 30x 2 - 12x + 1
a) a + b
b) a 2 + 2ab + b 2
c) a 3 + 3a 2b + 3ab 2 + b 3
d) a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4
Answers may vary, e.g., I noticed that the coefficients have a
symmetrical pattern. I also noticed that the coefficients can be
predicted using the following pyramid, where each number is the sum
of the two numbers above it:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
NEL
6706_PM10SB_Answers_pp556-567.qxd
5/13/09
1:14 PM
Page 563
Lesson 3.5, page 175
2.
a) Answers may vary, e.g.,
1
y = - (x - 2)(x - 4) = - 0.25x 2 + 1.5x - 2
4
b) Answers may vary, e.g., - 0.75
a) Answers may vary, e.g.,
y
6
a) 60 m
b) Answers may vary, e.g.,
y
80
60
40
20
-2
0
-2
2
4
6
7.
y = - 0.2(x + 1)(x - 5) = - 0.2x 2 + 0.8x + 1
b) Answers may vary, e.g., about 1.75 m
a)
8.
2
20
x
0
9.
10.
2 4
b) 0, 4.6
c) y = - 5x(x - 4.6)
d) maximum height: about 26.5 m
Answers may vary, e.g.,
a) y = - 5x 2 + 22.5x + 2
b) about 15.3 m
c) about 27.3 m at about 2.3 s
a) Yes. The second differences are the same: 12.4 cm.
b) 200 y
Glossary
4.
6
c) Answers may vary, e.g., y = - 5(x - 6)(x + 2)
d) Answers may vary, e.g., about 79 m
a) 10 m
b) Answers may vary, e.g., y = 1.2x 2 - 12x + 10
c) Answers may vary, e.g., -20 m at 5 s
a) Answers may vary, e.g.,
y
30
10
Answers may vary, e.g., y = 0.6x 2 - 0.6x - 3.6
b) Answers may vary, e.g., about 0.6
a) Answers may vary, e.g., y = 2(x + 3)(x - 1)
b) Answers may vary, e.g., y = 2x 2 + 4x - 6
c)
4
Appendix B
3.
x
x
0
-2
4
2
Appendix A
1.
6.
160
120
80
40
x
0
Answers may vary, e.g.,
a)
y
40
11.
12.
30
Answers
5.
1 2 3
c) Answers may vary, e.g., y = 24.8x 2
d) Answers may vary, e.g., about 1.8 s
e) Answers may vary, e.g., about 131.2 cm
1600; the number of dots in the diagram, n, is equal to (2n)2.
a), b)
20
10
0
x
10 20 30 40 50 60
NEL
Figure 4
x
y
1
3
2
8
3
15
4
24
5
35
Figure 5
First
Difference
Second
Difference
Index
5
b) y = x (x - 55)
126
c) horizontal distance: about 27.5 m; maximum height: about 30 m
d) about 24 m
5
7
9
11
2
2
2
Answers
563
6706_PM10SB_Answers_pp556-567.qxd
13.
14.
5/13/09
1:14 PM
c) y = (x + 1)2 - 1
d) ( -2, 0), (0, 0)
e) y = (- 2 + 1)2 - 1 = 0; y = (0 + 1)2 - 1 = 0
f ) The value of x must be 1 or greater.
No. If the curve does not cross the x-axis, then the factored form
cannot be used.
Answers may vary, e.g.,
Page 564
2.
3.
Increasing a makes the parabola narrower; increasing b shifts the
parabola down and to the left; increasing c shifts the parabola up.
a)
y
6
-4
2.
3.
4.
5.
(–5, 0)
a) 30
3 * 3 * 3 * 3
1
= = 1
b)
3 * 3 * 3 * 3
1
c) Any non-zero base with exponent 0 is equal to 1.
53
5 * 5 * 5
1
= = 1
d) 3 = 50 =
5 * 5 * 5
1
5
a) 3 - 1
3 * 3 * 3
1
b)
=
3 * 3 * 3 * 3
3
c) Any non-zero base with the exponent - 1 is equal to the
reciprocal of the base.
52
5 * 5
1
d)
= 5-2 =
= 2
5 * 5 * 5 * 5
5
54
52
5 * 5
1
e)
= 5-3 =
= 3
5 * 5 * 5 * 5 * 5
5
55
1
1
1
1
1
a)
b)
c) 1
d)
e)
f)
16
25
81
4
49
1
1
1
1
1
a) b) c) - 1 d) e)
f) 32
16
5
9
64
27
16
1
8
16
a)
b) 4
c)
d)
e) f)
27
8
9
9
4
7.
5-2 is greater. It will have a denominator that is less when it is
written in rational form.
( - 1)-101 is less. Since it has an odd exponent, it will equal -1.
In comparison, ( -1)-100 has an even exponent and will equal 1.
c) 0
d) 3
e) -2
f ) 2 or -2
Chapter 3 Review, page 185
a) No. It is a first degree, or linear, relation.
b) Yes. The second differences are constant and non-zero.
c) Yes. It is a second degree relation.
d) Yes. It is a symmetrical U shape.
564
Answers
6
0
-8 -4
-6
i) x = - 1
a) - 3
1.
ii) (4, -16)
iii) 0
iv) 0, 8
y
(3, 0)
x
4
y x2 2x 15
-12
(0, –15)
(–1, –16)
6.
8.
(4, –16)
i) x = 4
a) 449
b) Model 25
b) 3
x
(0, 8)
y x2 8x
b)
Use graphing
technology to estimate
values of a, adjusting
as necessary.
Lesson 3.6, page 181
1.
-6
-18
or
15.
4
-12
Calculate or estimate
the two zeros: r and s.
Substitute the coordinates
of a known point into
y a(x r)(x s) to
solve for a.
(0, 0)
0
ii) (- 1, -16)
iii) -15
b)
iv) -5, 3
4.
a)
5.
a) maximum, because the second differences are negative
b) positive, because it is between the zeros and the curve opens
downward
c) 1.5
Answers may vary, e.g.,
6.
x
y
1
2
2
7
3
14
4
23
5
34
x
y
1
1
2
7
3
17
4
31
5
49
First
Difference
Second
Difference
5
7
9
2
2
2
11
First
Difference
Second
Difference
6
10
14
4
4
4
18
NEL
6706_PM10SB_Answers_pp556-567.qxd
5/13/09
First
Difference
Second
Difference
y
2
2
7
3
11
4
14
5
16
Page 565
5
18.
-1
4
-1
3
-1
b) Yes. It is a symmetrical curve with a U shape.
c) y = - 5x 2 + 30.5x
d) about 36 m
e) about 0.7 s and about 5.4 s
Answers may vary, e.g.,
y
a), b)
1200
Appendix A
7.
x
1
1:14 PM
1000
2
800
Each table of values represents a parabola because the second
differences are constant and not zero.
a) i) 0, 18
ii) x = 9
iii) (9, 81)
600
400
200
x
0
b) i) 0, -2.5
ii) x = - 1.25
iii) ( -1.25, -9.375)
20.
c) Yes. The equation to describe the curve is degree 2.
d) y = - 5x 2 + 1200
e) about 15.5 s
25
1
1
a)
c)
e)
8
4
64
1
b) d) 1
f ) -36
5
1 2
3-2 is greater, for example, because it equals a b , which has a
3
1 2
denominator that is less than the denominator of a b .
4
Answers may vary, e.g., using graphing technology, I can see that x 2
is greater than 2x when x is between 2 and 4.
Chapter Self-Test, page 187
8.
9.
10.
11.
14.
15.
16.
17.
1.
2.
3.
4.
0
-8 -4
-8
(0, –12)
-16
5.
6.
40
4
8
(2, –16)
10
(–4, 0)
x
(6, 0)
0
-8 -4
-10
4
8
a) 51 600
b) Answers may vary, e.g., between 1974 and 1983
a) 10x 2 - 11x - 6
b) 15x 2 - 14xy - 8y 2
c) -5x 2 + 40x - 80
Index
30
zeros: - 6, 2; vertex: (- 2, -4); equation of the axis of symmetry:
x = -2
a) x = 5
1
b) y = (x + 9)(x - 19)
7
171
1
10
c) y = x 2 x 7
7
7
a) Yes. The second differences are constant.
b) Yes. The second differences are constant.
a) y (x 6)(x 2)
b) y (x 6)(x 4)
y
y
30
(0, 24) (1, 25)
8
20
(–2, 0) (6, 0) x
20
10
0
-10
NEL
x
4
8
Answers
Answers
12.
13.
Answers may vary, e.g., the greater the value of a, the narrower the
parabola is. Also, a positive a means that the parabola opens upward,
and a negative a means that the parabola opens downward.
a) either 2000 or 5000
b) 3500
a) y = 2(x + 2)(x - 7)
b) (2.5, - 40.5)
a) y = 0.5(x - 5)(x - 9)
d) y = 0.5(x - 4)2
b) y = - 0.16(x + 3)(x - 7)
e) y = - 4(x - 3)(x + 3)
c) y = 0.75(x + 6)(x - 2)
$2.00
a) 2x 2 + 3x - 9 = (x + 3)(2x - 3)
b) 15x 2 - 38x + 24 = (5x - 6)(3x - 4)
a) x 2 + 9x + 20
d) 12x 2 + 7x - 10
b) x 2 - 7x + 10
e) 20x 2 + 2xy - 6y 2
c) 4x 2 - 9
f ) 30x 2 + 32x - 14
a) 4x 2 + 24x + 36
c) 32x 3 - 2xy 2
b) 12x 2 - 14x - 40
y = –0.25x 2 + x + 8
Answers may vary, e.g.,
a) 50 y
Glossary
21.
8 12 16
Appendix B
19.
4
565
6706_PM10SB_Answers_pp556-567